It seems the MySQL "Connection refused" error is only being logged to the file-log, which is used when the database is inaccessible - makes sense. However, looking at these logs, the connection refused was last logged days ago, while I'm seeing very frequent errors running tasks, the last one happening today. If I look at the System logs in the Support section of the Admin CP, which are being logged to the database, I can see the error that is being reported - unrelated to the connection refused:
This is the most frequent error, and it is occurring multiple times per day. It seems to clearly be a PHP execution error within the IP.Board application source files.