You’re still just looking at a whole bunch of possibilities.
However, the difference between cold and hot is the difference between the engine operating in “open” vs. “closed” loop operation. During open loop the PCM runs a rich fuel mixture (the equivalent of what used to be the “choke” in the days of carburetors). After you get to operating temp it goes into closed loop which means it will be adjusting the fuel air mix based on info from the oxygen sensors.
One primary input will be the coolant temperature sensor and if it is bad it will leave you in open loop & leave the car running too rich. (A stuck open thermostat could did this too, but that would be accompanied by other things like poor heating inside the car). The other thing is that O2 sensor issues will throw off the PCM in terms of making the optimal fuel/air mix.
The thing is that the entire story can get complicated. So the engine temp issue suggests some possibility that you’re having this kind of problem. But it could be that you still just need new plugs or fuel filter etc.
Either way certainly give the mechanic the info.