P0420 + P0300 is it the chicken or the egg?

Hi gang. My wife's 2011 Prius had been running fine 221K on the clock. My son was driving it and he called me and said the check engine light was flashing I told him to pull over and he said then it went solid The car was misfiring it felt. I went to him and read the codes and had p0420 and then misfire codes p0300, 301, 302, 303. I also read the temperatures at the front and the back of the cat It was about 160° hotter at the front than at the back. It's my understanding it's supposed to be the other way around. I read somewhere that a bad cat cause misfires I'm assuming through back pressure. But could it be possible that the misfires sending unburnt fuel into the cat would clog it? It doesn't seem that's likely because why would it all happen at once. I am not sure which to address first. I've taken the cat off another Prius once and cleaned it and it worked fine for another year