I use a Huawei Mate 10 pro and the pre-installed (Android) email app. Unfortunately the emails are not synchronized with GMX. I get new mails on my smartphone, but if I delete them on my smartphone they won't be deleted on GMX. That would not be my "main problem" in and of itself. It is much more annoying that mails that I delete on GMX or move to another folder continue to appear in the app.
Is there a setting I have to make here? Or do I need another mail app (e.g. From GMX)?
So far I have avoided another app, because the pre-installed mail app can't be deleted or downloaded again and again.
Check whether you have subscribed to all folders. With IMAP accounts you may have to Subscribe to certain orders additionally because they are not included as standard.
The help function at GMX is sure to be a good guide for this
See that you set up the connection via IMAP.
That should give you at least technically the possibility to sort your e-mails into folders. (With POP3 I couldn't see any such function at first glance.)
Deleting emails should also be possible using POP3. If that doesn't work after the switch to IMAP, then you either have to change something in any settings of the app or get an app that offers this functionality.