It this your system?
If so, then I don't know if the power supply is replaceable. You'll need to scour the HP forums and tech support options to make sure. Even so, you're probably still going to be limited to a single slot width graphics card, given the configuration of your system. Additionally, you're going to be limited by the physical length of the card as it appears the CPU cooler would be in the way for longer graphic cards.
Given your situation, I'd first measure the space between the CPU cooler and the rear of the case to determine what length graphics card would fit. Then I'd check the length of various single slot (the GPU cooling heatsink/fan doesn't extend over to a second expansion slot) Radeon HD6450/HD6570 graphic card. Neither of these cards should require an power supply upgrade, but neither are going to be able to play games very well either.
Edit: Forgot to mention that if the linked system is your system, then in addition to being short in length and low powered, the graphics card also needs to be low-profile.