Auto BMW 520 Touring sometimes struggles to initially accelerate?

I have recently bought a 2000 Auto BMW 520 Touring with approx 95k mileage. I bought a CD Player through Halfords who fitted it and removed the BMW Business player. A week or so later, the Air Con wouldn't switch off even when the keys were not in the ignition eventually resulting in flattening the battery. After jumping the car with jump leads, I somehow managed to get the Air Con off but every now and again it would come back on and turn off at will; once more flattened the battery. I jump started the car again and that was the last time I had any problems with the air con but I think the battery is now so damaged that if I leave the radio or something on for anything more than 10 mins without the engine being on, the battery dies and needs to be jumped again. I have now learned my lesson and don't leave anything on without the engine being on and I intend to get a new battery in the next few weeks. However, my car has now developed a new problem which I am hoping is linked to having a duff battery. Every now and again (but becoming more regular), when I start the car, it can initially really struggle to accelerate for about 2-3mins but after that it runs fine? Will this be rectified if I buy a new battery or is it likely to be another problem?