Fritia - Hush

If you don't have Hush yet but want to try her out, she is available as a Trial Operative in all of the missions in Chapter 11 "Fogbound Dream"
(Main Story // Legacy // Containment Zone - Endless Snow // Chapter 11 - Fogbound Dream)
She is also available to try in Story Missions 8-10, 8-11, 8-12, 9-9, SP3-7, and SP3-8

With plenty of investment (notably M3, as well as Tess - The Magician as a support) Hush is a fun and competent operative, dealing massive damage from her ult, however without high investment Hush is extremely weak - infrequently using short medium-damage ults. Regardless, whilst her ult is fun, she spends most of her time in combat ineffectually shooting her gun for little effect as she waits for her U-Energy to recharge, which can also make her feel very slow to play.

Hush does have a naturally large HP pool and often gives herself some damage resistance, which can give her good survivability, although her ults low mobility means she needs it.

Due to her massively benefiting from higher-investment, Hush is generally an all-or-nothing operative - either go all-in on her, or don't bother with her.

The Basics

Role: Main DPS
Deals Thermal Damage
Weapon Type: Assault Rifle
Deals almost entirely Skill Damage
Mobility
During Ult: Literally None
Otherwise: Good

How to use Fritia - Hush

If at max U-Energy, then use Hush's ult
Otherwise, shoot stuff and spam her standard skill off-cooldown.

How to build Fritia - Hush

Weapon: Lava Bones / Warhammer
Weapon Parts: Max ATK
Logistics Squad: Thebes
Basic: +10.0% ATK
+106 Alignment Index
Special: +8.5% Thermal Damage
Best Supports:
Vidya - Celestial Swan
Siris - Freefin
Tess - The Magician
(If one ult is enough to kill the enemy)
Marian - Riptide

Specifics:

Ultimate Skill

Almost all of Hush's damage comes from her Ult.

Hush' ult causes Hush to fire her weapons platform, which is a thin beam weapon that deals Thermal Damage.
This beam deals damage every 0.1 seconds.

The damage increases the longer the ult is used for - 2% per 0.1 seconds (or per hit, since it hits every 0.1s).
I.e. After 1 hit it's 2%, after 2 hits it's 4%, after 3 hits it's 6%, etc.
This is an additive bonus. Given how many hits her ult can do, this can stack up to very large amounts - at 15s Hush's ult is doing 400% of the damage it started with.

Duration

Hush's M1 allows her ult to last for as long as she has U-Energy.
If Hush has M1, then the increasing damage bonus means Hush should only ever ult at max U-Energy for max ult duration and thus max damage.

At M3, Hush gains U-Energy as she unleashes her ult, gaining 0.3 U-Energy per hit on a burning enemy, so +3 U-Energy per second.
This can extend her ults duration for more damage.

Hush increases the squad's Max U-Energy by 20.
Hush also gains reduced U-Energy Consumption, with extra U-Energy consumption Reduction from her Alignment Index.
Both of these help with her ult's duration, and thus damage.

Mobility and Survivability

Whilst firing her ult, Hush can only cower from the blastback of her beam, rendering her completely immobile.
Hush can still change the direction the beam is pointing in though.
Hush gains a shield equal to 20% of her Max HP when using her ult.
Hush does gain quite a few damage reduction mods, and has a naturally huge HP bar, but being completely immobile during her ult is still very bad for her survivability.

Standard Skill

Hush throws a projectile at the enemy, which pulls enemies in, applies a Burn (Thermal DoT), stuns enemies, and deals AoE thermal damage.

This skill groups enemies quite well, and does inflict burn which is useful, but its dps is very poor, and Hush can't really make much use of enemies being grouped up anyway as her gun and ult are both single-target attacks.
Overall, this skill is extremely weak.

Support Skill

Hush gives your deployed operative 5 stacks of Fire God Protection, which reduces the damage from the next hit and burns nearby enemies.
This burn deals a total of 100% of Hush's ATK per application.

Whilst certainly not the worst support skill on a main DPS, you should never be using Hush as a support operative - she's a main-DPS only.

Weapon

Lava Bones
Her 5* weapon, and her best weapon.

The game recommends Strawberry Shortcake as her 4*, but Strawberry Shortcake is only useful for support operatives, which Hush isn't - she's a main DPS only.
As such, if you don't have Lava Bones then you should try:

Warhammer - Just a straight +Thermal damage buff.
or
Foul Butcher - A 5s damage increase to her ult, but it's a pretty good increase.
Remember that Hush herself increases your squads U-Energy cap, so the max bonus Foul Butcher gives isn't 30% Skill DMG - it's 36%.
If you have Hush at M1, then her ult can last over 5s, meaning the bonus can be-reapplied as Hush gains U-Energy.
If you have Hush at M3, then Hush can gain U-Energy from her ult, proc'ing the weapons effect again for more damage.

Either weapon is fine. I'd generally recommend Warhammer over Foul Butcher as it's easier to use, however you may already have gotten a Lvl.50 (or higher) Foul Butcher in order to get its Flash Suppressor, in which case you may want to consider using that until you get Lava Bones, rather than spending extra resources upgrading Warhammer, as neither of them are that good, and Hush really wants Lava Bones.

For weapon Parts Hush wants max ATK.

Logistics

Stats

For stats Hush wants +10.0% ATK and +106 Alignment Index.
For the Special talent, Hush wants +8.5% Thermal Damage

Alignment gives her reduced U-Energy consumption, which is very useful.
Skills can't crit, and most of Hush's damage comes from her ult, so crit mods are not good for Hush.
Skill haste does nothing, as her standard skill is basically useless and her ult is entirely limited by U-Energy.
U-Energy recovery is a decent stat, but Hush generally just prefers the U-Energy cost reduction from her Alignment Index. U-Energy Recovery is a good cope stat though.

Squad

As for squads, Hush has three real choices:

Thebes squad - Skill DMG +24%. Each time a Standard, Ultimate, or Support Skill hits the target, increases own ATK by 2% for 5s sec. Max 20 stacks.
At max stacks, Thebes gives +40% ATK. In order to maintain this bonus, you need to be doing damage every 0.25 seconds, which Hush's ult can easily sustain.
Hush' ult deals damage every 0.1s, so Hush's ult takes 2 seconds to max the buff.
Fenya squad - Ultimate DMG +32%. Increases DMG dealt by 46% for 10 sec after using Ultimate skill
Fenya gives Hush's Ults +78% damage for the first 10 seconds, and +32% afterwards.
You can get Hush's Ults to over 10s duration, but this does require some investment.
Xinye squad - +32 max full-squad U-Energy [stacking]. Full squad U-Energy Recovery +28%. Using an ult recovers 20% of the U-Energy used.
Having more max U-Energy means you can sustain Hush's ultimate longer, for more damage, and more regen means you can ult more often.
This can allow you to spam out ults much more often, although each one will deal less damage than if you had used Fenya or Thebes squad.

Without M1, Hush' ults are limited to 4s in duration, making Fenya or Xinye the best choice depending on whether you want more powerful or more frequent ults, respectively.

With M1, Hush's ults can last over 4s.
Hush's ult does increasing damage over time, meaning that the final bit of Hush's ult deals the most damage, but that's also the part of her ult that will not be buffed by Fenya squad's bonus as Fenya only lasts 10s.
Whilst Fenya does have a larger number, Thebes not having a duration cap means it works out better for longer ults.
Similarly, Xinye's shorter-duration ults deal quite a lot less damage due to Hush's ults dealing more damage for later damage ticks, so even though Xinye allows Hush to ult more often, the lower damage per ult means Xinye is not as good as Thebes.

Since you'll get Hush to M1 eventually it's better to just get Thebes to begin with, rather than investing into Fenya or Xinye and then needing to change later.

Manifestation

M1 allows Hush to unleash her Ult as long as she has U-Energy, but the U-Energy cost increases over time.
Specifically, the U-Energy cost after 4 seconds is (11+4t) (where t is the duration in seconds over the initial 4)
So 15 U-Energy at 5 seconds, 19 after 6 seconds, 23 after 7 seconds, etc.
The cumulative total U-Energy cost is 15 after 5s, 34 after 6s, 57 after 7s, then 84, 115, 150, 189, 232, 279, 330, 385, 444, 507, 574, 625, and 720 after 20s.
With M1, Hush's damage goes up by a truly vast amount, but you will want to get her to max U-Energy before using her Ult in order to maximize damage.
M2 gives Hush a stack of Fire God's Protection every 0.5s whilst using her Ult.
This reduces the damage Hush takes by 50%, and burns the attacker. A stack is removed each time Hush is hit.
Since her M3 wants her Ult to hit a burning target, this is useful for keeping your target burned.
Offensively, M2 is not that useful without M3.
M3 allows Hush to recover 0.3 U-Energy every time her ult hits a burning target.
This should ideally be every hit, meaning Hush recovers 0.3 U-Energy every 0.1 second (as that's how often her ult hits).
This is a huge amount of U-Energy recovery, and it increases the max sustained duration she can use her ult for very dramatically.
M3 is thus a huge boost for Hush's damage.
M4 is a damage bonus to her ult.
This is good, but it doesn't change how Hush is played - it's just more damage.
M5 gives extra damage reduction for each stack of Fire God Protection.
Combined with M2, this is actually a huge survivability boost.
...that Hush really doesn't need.
M5 doesn't boost damage, and Hush is already quite bulky without M5 anyway, so M5 really isn't overly useful.

If you actually want to use Hush, you should get Hush to M3.
M4 adds some damage, but it doesn't change how she plays.
M5 deals identical damage to M4, and thus is unnecessary.

As such, if you want to use Hush then stop Manifestations at M3, unless you want her as your primary DPS in which case go M4.

Since Hush is part of the standard gacha banner, you'll max her eventually, so it's not worth pulling for her specifically unless you find her extremely fun.

Good Supports for Fritia - Hush

Vidya - Celestial Swan (and other resistance shredders)
Celestial Swan and other resistance shredders (Acacia - Kaguya, etc.) that reduce the enemies resistance help enemies take more damage from Hush's ult.
If using Celestial Swan, make sure the other support isn't off element to Hush.
Siris - Freefin
Freefin provides strong skill damage buffs, and Hush can make use of Freefin's Wavehunt thermal skill damage buff, although she can't maintain it.
Having to re-use Freefin's support skill every time before Fritia ults can get annoying, but Freefin is an amazing support for Hush.
Tess - The Magician
Tess' U-Energy recovery buffs are a huge bonus for Hush, as they allow her to use her ult more frequently, and for longer which boosts damage.
Tess allows Hush to do one ult immediately. If Hush can kill the enemy in one ult, then Tess is Hush' best support.

Eatchel - The Cub
Eatchel's ATK boost is always useful.
Hush being immobile during her ult means Fritia can potentially appreciate Eatchel's healing as well.
Marian - Riptide
Riptide's Skill DMG boosts are very nice, although her M2's 8s duration makes her difficult to use effectively with Hush's ult, which can last for well over 8s.
If you can't time it well, then Hush's ult just won't get Marian's full M2 bonus, which will make Riptide a mediocre support overall.
Overall Riptide is good, but you really want to make sure that you are timing Hush's ult so that Riptide can get Tidesong part-way through Hush' ult.
Chenxing - The Observer
Chenxing's Skill DMG buffs are very useful for Hush, and Chenxing's 15s buff duration allows it to last for most/all of Hush's ult.
Chenxing's healing is also useful for Hush during her ult, where she is immobile.
That Chenxing's support skills buff is a PBAoE isn't an issue for Hush, as Hush is immobile during her ult.