I would recommend that you spend a little time on suitable web sites and viewing a range of cars so that you know how to spot something that wont break down in a week. Spending a lot of money alone wont remove that possibility. My advice would be the same whether or not you had mentioned past problems and that is to avoid car finance. If you save your £500s for a few months and go shopping with that, you will have enough to get something decent.
It's nearly 3 years since I changed my car. I splashed out £300 and it's still going strong in spite of being abused although I was disappointed to get a £200 bill for the last MOT but had I been in better health I would have done more myself. Deals like that don't come along every day but, as I said, it wont take long to save up for something decent with £500 to spare each month. The hard part is learning to distinguish between good cars and worn out ones but a good starting place would be to avoid trade sales and only consider buying when the current owner seems decent and has been using the car for a while. It's worth looking at www.autotrader.co.uk each week.