Hesitation on acceleration

The engine starts fine, and runs perfectly for the first two stops. On the third stop and sometime the fourth, there is a hesitation when trying to accelerate from the stop. After that, it runs perfectly. A couple of times, the engine shut off, but started up again.

Six months ago, in winter, the engine would shut off, and it took several minutes before I could get it started again, then it ran perfectly with no hesitation.

The shop found low fuel pressure, replaced what I think was the ignition module, and the computer. It worked fine for a while, then the hesitation has started again.

Coincidentally, the heater stops working when the engine has the hesitation problem.

My mechanic and I are having fits trying to figure this out.