It is very hard for a home user to verify a motherboard. There are just two many parts that could go bad with no or little visible sign.
Do you know how old the Desktop is? Before 2006, more or less, there is not much you can do to upgrade. After that year some significant changes were made in desktop designs. Mostly for better.
"A few of the ready made Desktops were fair game machines with a little work."
- From the comments of others. I don't really know of a standard desktop was ever a good game machine.
If you have a very modest budget, you might think about a used console instead of a PC. See if any of your friends will trad you the HP for a console.