Backfiring ans sputtering when accelerating

Hello. It seems your PT Cruiser might have low fuel pump pressure. This will cause a backfire/misfire condition upon 'mild' acceleration. 'Gentle' acceleration, your car will run 'decent'. Good luck and keep me informed; if you want to.

What was the purpose of replacing the pump ? And what year is the Dodge ? And did they also replace the fuel filter ?

spark plugs dist cap spark plug wires

You can check the compression and if it OK then it may be a plugged fuel filter or bad electric fuel pump. The pump is in the gas tank.

How many miles? I am NOT a mechanic! Sounds like possible timing belt/chain jumped a tooth. Or the timing advance mechanism is malfuctioning. (Unlikely - check distributor is tight or not shifted from normal position too far back or forward.) Assuming belt and not chain, 80 k miles or more, likely need to replace timing belt in any case, before it breaks. If it breaks, you'll be replacing valves and pistons too.

As you have replaced all the obvious things that could cause the symptoms it may be a good idea to have a cylinder compression test done , if comes back OK i'd try a fuel addative , one that covers the catalytic convertor , as there are many sensor's involved in the fuel/exhaust system as you know , this addative will clean any residue affecting there reading's , hope this helps

Mine had the same problem it was the fuel pump.

it sounds more like the fuel pump itself

not positive but in the air breather there is a mass air flow sensor and sometimes air filter particles gets stuck on the sensor.It will look like a white piece of lint,if there is be careful to remove it and try it.Usually a backfire and sputtering is the signs

is the engine lite on? if a misfire shows, and the plugs are new, could be coil or wires. how does it idle? maybe air leak between mass air flow sensor and throttlebody.