Car Trouble Starting

[Joji]

Hi,
My Water pump broke last week and the belt fell off cutting the oil line and the coolant and oil all spilled all over the highway and then the car came to a halt and overheated and died. My mechanic fixed it. When I left the shop it started up fine. I drove a half hour home. the car sat for about a half hour. When I tried to go somewhere the car had trouble turning over. I turned the key and it sounded like it was barely chuggin along to start the card. it took about 30 seconds before it reved up. If the the car has been sitting for a while like over night it usually starts up fine but I do notice that when it turns over it rumble for a couple of seconds kind of like trying to catch its breath . When driving there is no problems. never stalls or anything.
Before I bring it back to the mechanic. Is it possible that the Battery is the problem or does it sound like it could be the starter. my mechanic said that when the car over heated after the Water pump broke that it might have melted or damaged parts of the starter. Please help because. The battery is only a year old. I can't tell if I am getting scammed to pay for more repairs.
thanks