5.7 hemi rough idle and low backfire out exhaust any ideas

One cylinder on regular misfire, or random misfires. Hot or cold the same?
I bought a used Charger where No 2 exhaust was not spinning and seems to have the seat and face carbon caked.
It's been spinning now for a few months and is slowly getting better.
When I first got it, it would always start and run and idle smoothly, but after the first time past 1600rpm the valve wouldn't seat right.
Thump, thump, thump out the pass side exhaust.
Shut off and restart and perfect again.
Now it's funny while cold, but at full hot almost perfect. I'm interested in hearing more about your situation.