Start with updating to the latest release of MediaMonkey. Then check if the files are not read-only and if MediaMonkey has rights to modify the files.

Many tracks and albums in my library will not save album art either to the library or to the track itself. All tracks in my library are either mp3 or flac and almost all are encoded with mediamonkey. There doesn't seem to be anything consistent with which files accept artwork and which don't other than the fact that only the ones that are encoded/ripped with mediamonkey have the missing artwork issue. Deleting the tags and retagging from the web doesn't fix the issue.

Running Mediamonkey 4.1.11.1783
Windows 10