April's GOG game is Wolfenstein: The New Order:
https://videogames.si.com/news/amazon-prime-gaming-april-2023-free-games
Edit: The Beast Inside as well:
https://worthplaying.com/article/2023/3/30/news/136533-prime-gaming-april-2023-free-games-loot-lineup-announced/

Prime Gaming for April. Store, games and dates.

GOG.
Wolfenstein: The New Order (April 6)
The Beast Inside (April 13)

Legacy app.
Looking for Aliens (April 27)

Amazon app.
Ninja Commando (April 6)
Art of Fighting 3 (April 6)
Crossed Swords (April 13)
Ghost Pilots (April 13)
Icewind Dale: Enhanced Edition (April 13)
Beholder 2 (April 20)
Metal Slug 4 (April 20)
Ninja Masters (April 20)
Terraformers (April 20)
Grime (April 27)
Magician Lord (April 27)
Sengoku (April 27)

And members in United States, Germany, Canada and the United Kingdom can play for free on Amazon Luna the rotating selection of games: Horizon Chase Turbo, Yakuza Kiwami 2, The Adventure Pals and The Jackbox Games Party Pack 3.

Wolfenstein: The New Order now up!
https://gaming.amazon.com/loot/wolfensteintno

Prime members have the ability to claim Wolfenstein: The New Order from April 6, 2023 to May 3, 2023.
Before June 2, 2023, redeem your offer code on GOG.com.

Prime Gaming for May :

May 4th :

- STAR WARS™: Rogue Squadron 3D [Amazon Games App]
- Super Sidekicks [Amazon Games App]
- Samurai Shodown IV [Amazon Games App]

May 11th :

- Planescape: Torment: Enhanced Edition [Amazon Games App]
- Lake [Amazon Games App]
- Robo Army [Amazon Games App]
- Last Resort [Amazon Games App]

May 18th :

- Kardboard Kings [Amazon Games App]
- The Almost Gone [Amazon Games App]
- 3 Count Bout [Amazon Games App]
- Alpha Mission 2 [Amazon Games App]

May 25th :

- Lila’s Sky Ark [Legacy Games Code]
- Agatha Knife [Amazon Games App]
- King of the Monsters 2 [Amazon Games App]
- Kizuna Encounter [Amazon Games App]

But there is a GOG game: https://www.gog.com/en/game/the_beast_inside

The Beast Inside was given on Prime in April although it might still be claimable for awhile.

Huh?
I must have missed it.
I just saw it today...

Edit: ah yes...I see:
https://www.gog.com/forum/general/amazon_prime_games_and_drm/post93

Fun fact: I did claim "Wolfenstein" - and "The Beast Inside" was nowhere to be seen at that time.

Edit 2: doh,...of course it wasn't - it came a week later. XD

Beasts of Maravilla Island [Amazon Games App]
Calico [Amazon Games App]
Double Kick Heroes [Amazon Games App]
Shattered: Tale of the Forgotten King [Amazon Games App]
Tandem: A Tale of Shadows [Amazon Games App]
Tiny Robots Recharged [Amazon Games App]
Turnip Boy Commits Tax Evasion [Amazon Games App]
DKO: Divine Knockout [Epic Games Store]

A reminder. As post95 announced, another 4 games are available to claim:
-Lila's Sky Ark. Ends in 188 days. Legacy Games Code.
-Agatha Knife. Ends in 33 days. Amazon Games App.
-King of the Monsters 2. Ends in 188 days. Amazon Games App.
-Kizuna Encounter: Super Tag Battle. Ends in 188 days. Amazon Games App.

Probably won't be an issue for most but I thought I'd say those NeoGeo games will require Windows 10. Doesn't really make sense to me since they've been playable in DOS for about 25 years.

Prime Gaming for June. Store, dates and games.

Amazon app.
June 1.
Mutation Nation.
Sengoku 2.

June 8.
Over Top.
Soccer Brawl.

June 15.
SteamWorld Dig 2.
The Super Spy.
Top Hunter.

June 22.
Autonauts.
Neverwinter Nights: Enhanced Edition.
Revita.

June 29.
Once Upon a Jester.
Roguebook.

Legacy app.
June 29.
Gems of Destiny: Homeless Dwarf.

A little aside:
Out of boredom, I gave that "Amazon Luna" streaming service a chance for the first time yesterday and played RE 2.
I was surprised how well it worked.
I honestly wasn't able to notice any difference between playing via that streaming service and playing a copy installed on my PC.
(though I have to say I'm using LAN - not WLAN on my desktop PC)

Today, the four new "free to play" games for June have been made available:
- Batman: Arkham Knight https://www.gog.com/en/game/batman_arkham_knight
- Citizen Sleeper https://www.gog.com/en/game/citizen_sleeper
- SteamWorld Heist https://www.gog.com/en/game/steamworld_heist
- Monster Harvest (n.a. on GOG)

Neverwinter Nights: Enhanced Edition is available as a key redeemable on GOG as part of Amazon Prime’s subscription.

It surprised me given that (1) the Prime Gaming blog mentioned it starting on the 22th for the Amazon Games app and (2) every prior Beamdog game on Prime has been an Amazon Games claim.

PS: the code activates a new package and doesn't check for game ownership (just like the "Digital Deluxe" that refuses to be marked as owned in the store. Double check in case you already own the game.