Purchased used car. One month later it begins misfiring randomly.

Code is p0300. After cold start and sitting idling. -
Check engine light comes on almost immediately upon start and begins flashing about one to three min into operation. Car shakes noticeably. Rpm’s surge slightly - you can hear it. Initially, a second repetitive growling vibration started and within 10 seconds, car dies. After trying a few things, it stopped doing the growling thing.
The things I tried that seemed to help were
Cleaning the throttle body
Removing air filter
Banging on catalytic converter.
Thoughts are... Air/fuel mixture isn’t right.

Background- initially another person changed spark plugs, added gas treatments, Had compression tested. 1 and 2 are ok. Cylinders 3 and 4, respectively, are the trouble. Maf and MAP sensors and head gasket were changed. Mechanic suggested (signs of overheating were obvious and he couldn’t diagnose it) and so we replaced the ECM. Now Crankshaft Poaition Sensor has to be relearned- appt is made to do that. Air filter replaced with new. Mechanic says injectors clicked. Unsure how he tested them. I’m tracing wires and looking for continuity from each ignition coil connector to ECM because wiring harness in ignition coil area were sticking to each other after an apparent overheat.
So far nothing open or shorted.
Worth noting- inside of air intake cover (before air filter) has coating of oil inside it. Was told that when head gasket was replaced the inside was caked with “gunk”.

So still have a bad misfire on idle. No misfires seem to happen at road speed according to live data on Foxwell code reader.
When I removed the air filter, acceleration was snappy. That’s why I bought a new one. Is it possible the. Catalytic converter needs to be replaced? Mechanic didn’t find anything obviously wrong. - Suggested ecm my have been damaged at overheat. No change.