Car cranks over no start

So I just bought it with a big hole in one of the hoses. The person I bought it from did not know much about cars and said the car overheated (due to the giant hole). He tried to fix it himself and got the water pump replaced and got a new timing belt on it, so it could not be those things. I got the hose replaced and then tried to start the car. At first, it just turned over, but then I learned that if I stamped on the gas a bunch, it would slowly turn on and get moving. When it was moving, it would sometimes never go above a certain speed even though I was flooring it but it only got going about twenty miles per hour. It then would rectify itself and get going. One day, I was driving it on my 4th test and it randomly just turned off and would not get started again and I do not know why (I had to push it a block into my driveway). It still cranks over but it does not start no matter what I do anymore.