The first thing I would do is restart both PCs and see if the problem goes away. If yes then I would put it down to Windows idiosyncrasies or silent updates. If no, move on.
Then if not I would open Windows Explorer and check that I can view the F: drive successfully. If not, again, blame Windows networking. If yes, move on.
Finally I would right click on one of the "bad" file on the MC Client, and select "Locate>On Disk (External)" and see what happens. With my installation, with no shared mapped drives, I get an error saying MC can't find the file 'm01p://192.168.0.14:52199/MCWS/v1/GetFile?File=xxxx' does not exist.
See what you get, and compare that to what you get for a known "good" file that plays successfully.
If none of that solves the mystery, I would delete the local reference to the server on the Client, then delete the actual local copy of the library files, then add the server back in, forcing a full resync, and hopefully fixing the issue. Thumbnails will need to be rebuilt locally, but everything else should be there right away. That sounds drastic, but as the error message was "The database entries for the selected files are missing or invalid", it could be there is some corruption in the local copy of the database that isn't being fixed by the synchronisation process.