My daughter has a 2000 neon and when she pushes the gas peddle the car is not accelerating. Also the engine light comes on sometimes and then goes out..

Sounds like a basic miss or fuel problem.
Have you had the computer scanned for trouble codes ?

It could be your fuel pump

did you check and see if the cable snapped for the accelerator, have someone push the peddle a few times and you watch to see if the accelerator lever that the cables connect to move if not they obviously broke and need to be replaced

I have seen fuel pressure issues cause this. Fuel filter, or fuel pump. However, I have also seen the catalytic converter cause this issue. The honeycomb, located inside the catalytic converter, breaks up and will block the exhaust when you accelerate. Engine power is lost, until you let off the gas and let it idle. Then, the chunk falls down from the exhaust port, and it runs fine again. Until you get on the gas and increase the velocity of the exhaust, and the chunk blocks the exhaust gasses again. Good luck

with smoke mean time to replace engine valve is bad and oil is mixing with coolant internally

Most chain auto parts stores will scan the codes free or loan you a scanner to use on site. Scan the code (even if the light has gone off, it maybe held in the memory) and see what it comes back with. Post the code number here or Google it

check the accelerator .sound like it come out the holding bracket.

loose fan belt?

Put the car in neutral, set the parking brake, open the hood, get someone to listen/see where the sound is coming from, as you press on the accelerator.

Your throttle speed sensor(idle speed sensor)is not working properly.It's job is to keep the the engine at the proper rpm range when you are not applying the gas peddle.This is the most likely cause.

check fuel strainer