[xavierx]

If the rough startup is happening after the car has sat for a number of hours, and does the same when it is not cold, then it is possibly injector wear.

When the piston inside an injector wears to the point that it is no longer able to fully seal, it can cause the injector to slowly drip fuel into the cylinder over time.

When you next come to start the car, this causes two things 1) excess fuel in the cylinder causing a missfire and 2) loss of pressure between the high-pressure pump and the injectors, meaning it will take the pump a few seconds to get up to normal operating pressure.