I have been using the 'Mail for Gmail' app on my Mac for years without any issues. Just recently it has stopped working. Getting a blank screen. MacOS 13.1
Mail for Gmail app is a third-party app, and it's free. Those two things mean you're at the mercy of that developer.
The first thing I would try would be to uninstall the app, making sure to get ALL the preference files (which don't always get deleted automatically when you simply delete a Mac App Store app). Then re-install and see if that fixes the problem. Next thing to try is contacting the developer with the issue.
But, as @Sheepish-Lord stated, the app is 4+ years old – so it's highly likely that the app simply isn't going to work with macOS 13 and later – and the developer is going to say "sorry, the app is no longer supported."