I doubt it is the winding's. Winding's are usually shorted or opened on failure so the motor will trip a break or won't run at all.

If the pump is turned on but after a few minutes shuts off on it's own, then it is likely the thermal limiter switch tripping. This is caused by overheating in the motor which in turn is usually caused by bad bearings in the motor or even a problem with the impeller. Under voltage and over voltage can cause the problem as well but that is pretty rare. If you are comfortable checking the motor voltage, you should probably do that first just to make sure.

Is the motor loud when running?

How old is the motor?