I can mark it as SOLVED.
I reall don't have any clue, how's it possible, but the original Cachly folder with 2 subfolders Photos and Photos 2 was really deleted.
Even though I manually created a Cachly folder and a Photos subfolder, adding a photo to a private note didn't work.
Only when I restored the completely mysteriously deleted original folder created by the app and all the files in it in iCloud, adding a photo works again.
I'm not a lame when it comes to iCloud and the whole Apple ecosystem as well as the digital world and life, but I really can't explain this in any way. Simply "X-Files."
Conclusion:
a) problem solved
b) thank you for your efforts and willingness to help.