I just bought a polo, drove off very excited only to realise the battery is a dud, no service history etc, feeling like I bought a dud, trying to teach myself as much I can

My car keeps stopping every 200 clicks for no reason, but this time I have found that the key? is getting hot should that happen. Its a 1.8t 010 mod I love this car but it is doing my head in, and I wont sell it because I don't want to pass a dud on to someone. Who bought it in good faith off me.