I think I got the same problem.
Qt WebEngine Plugins directory not found. Trying fallback directory... Plugins as for example video codecs MAY NOT work.
Error in finalize query "attempt to write a readonly database" : "BEGIN EXCLUSIVE"
Error in open database "unable to open database file" : "/home/user/.local/share/data/Mendeley Ltd./Mendeley Desktop/www.mendeley.com/user@mail.com-f453/search-index.sqlite"
Failed to open search index
Error in prepare query "unable to open database file" : "SELECT value FROM UpdateTokens WHERE key = :key"
Error in bind value "no statement prepared"
Error in execute query "query not prepared"
Error in step query "query not active"
QObject::disconnect: Unexpected null parameter
The problem, I think, was that the folder where my mendeley database is stored changed its name (don't really know why).
To solve I just erased the folder /home/user/.local/share/Mendeley Ltd. and start mendeleydesktop again so the folder is created again (your papers will have to be sync again).