I see on settings page that apps such Twitter and Translate store some App Data:
How do I clean that App Data? Tried to clean Temporary files in Storage Check, but it didn't help.
In most cases, you can't.
App Data is simply the data the app stores while in use. In the case of Twitter, it stores/caches the avatars so it doesn't have to re-download them each time you use the app. Even though they are just small images, when you've viewed many thousands of avatars, that all ads up.
If an app doesn't have an option to clear this kind of data, then there isn't much you can unless you delete the app and re-install it.