When the car is warm after a while it is ...

Kinda sounds like the ECM or "start mixture" control module is faulty or is getting a bad input from a sensor. For example, it man not be getting a proper temperature sense from the exhaust manifold or the intake manifold water temp sensor. So, every start seems like a cold start....

If your cold start is on a "cold" engine - fine. If your cold start is on a "warm" engine - problem. Mixture is too rich for a warm engine.

Again, could be sensor sending a bad temp signal... or the control module not responding to the signal.

A bit too complicated a problem to exactly troubleshoot via long distance... but you now have a starting point.... -Jim