Hello!
That's exactly what was missing - thank you very much for your support!
Again an example of the original problem:
When I'm using an app - in my case a photo editor, I've edited a photo
and then wanted to save, came a dialogue that invited me
the location (destination) to determine or grant access. After clicking on "ok", the dialogue disappeared and you could not do anything else. Now comes the appropriate file manager and I can determine the destination (my HDD). With many other apps, it now works properly!
Thank you!