First of all 5650 is a gpu made for laptops.Any gpu will work in your system but you have to upgrade your psu to a decent 500watt.Who told you that you cannot install a psu larger than 350watt?
I am aware it is a GPU designed for laptops, but I'm not sure what to look for GPU wise. Which is why I'm here (heh). As for the information I recieved about the motherboard, that was from HP Tech support. When I posed the same question to them. They suggested a GT240, and told me I could only upgrade my power supply to a max of 350W (I intended to buy a Radeon 5750). I was confused by this and asked if it was the PCs housing or otherwise that limited the size of the power supply, and she told me it was my motherboard. Is this information incorrect? Thank you for your assistance.