Is there a way to make iPhone forget the data belonging to one app?
If you delete the app, all the data from that app gets deleted.

That is how the phone forgets that app.

If the app uses Game Center or another log in to retrieve game state from servers online, then you'll need to temporarily sign out of Game Center and possibly iCloud to remove those files from the phone as well.

I'd start by signing out of Game Center and delete/reinstall the game.

If that doesn't work, you can either check the specific game / contact that developer and/or sign out of iCloud documents or inspect that data store to see if that game has cloud files.

Once you have a clean start, you will know how to force a change to the cloud to reset the progress to be new.