I don't use caching plugins, I've found they cause more trouble than any bit of site performance improvement they might provide.
I have several sites on the same server, half running Powerpress, the other half still serving up legacy podcasts with podPress.
Two of the podpress sites have over 200 episodes in them, and the Powerpress sites have feeds of 100 and 50 episodes in them.
The podpress sites don't generate this mysql error, have never thrown this error, while the powerpress sites do, without fail.
This error has followed along even after moving the sites collectively from one hosting provider to another, even hosts with ridiculous amounts of resources allocated (12Gb RAM, both PHP and MysQL set to equally ridiculous timeout levels), and this error still appears on only the powerpress sites and never on the podpress sites.
Logic would indicate that your assumption about the problem being with the mysql server is incorrect, based on the same error appearing even after a site has moved providers, and with higher resources available.
(yes, I've been having this problem since I switched several sites from podpress to powerpress at your recommendation several years ago, and you know about my extensive background in unix administration and internet protocol troubleshooting. Remember all those FarPoint Media shows?).
The only reason I haven't switched any of those sites back to podpress to see if I'm right and get rid of the error is that I don't have the time.
But if you truly have absolutely no interest in digging into this on the powerpress side despite the avalanche of evidence strongly suggesting that it's the source means that maybe I should take the time.
Yes, that's my frustration speaking out loud.