Why would the engine while accelerating start choking itself out and then regain power when you turn the car off and on?

Sounds like fuel flow is limited. Check the fuel filter. This is a cheap replacement, and needs to be done every 50 to 100K miles anyway. If this doesn't help, check the fuel pump. If this doesn't help, look for obstructions to the exhaust. You see this in older cars where the exhaust system is not stainless steel.