why you cannot create new folders / edit file name / save files under your mac desktop?
The root cause of it is because your desktop is locked.

if you can create new folders / edit file name / save files in other folder, then you can do it in this way to solve the issue:
1. open the hard disk and right click the folder “Users”
2. select File -> Get Info, scroll down to “Ownership & permissions”
3. change Owner Access to “Read & Write”, change Group Access to “Read only” if it is not
4. navigate into “Users” folder, right click “Desktop” folder.
5. change Owner Access to “Read & Write” if it is not