My 4 cylinder 2.3 lt 4 door ex '02 accord has a sub par battery, but the other day we had an ice storm and a normal jump didn't help, when a large truck with large Gage cables came it started but only after I gave it gas while holding the key over, there was a kind of loud bang/puff sound but I was watching for smoke and nothing, car just started, and now days later it's like dead battery every time but if I press down the gas then starts right up, and runs great. Ever heard of this? I'm not very well off so I can't just replace the fuel pump or whatever just to see if that's it. Any diagnosis tips would be just as good as fixing tips I believe. Anyway, thank you all for your time

 

Also, I forgot, my cadalitic converter was stolen and I replaced it with a straight pipe. If that could matter.