This failure could mean a lot of things. Most power supplies use more than one connection for say the supply and the grounds. If the PSU was cheaply made they could have cut corners. I don't think I can write enough here to explain this but I'd start with a new quality PSU that is a) Single Rail and b) 50 to 100 more Watts.
-> In the future supply make and model numbers on card and PSU so I have more to work with.
When the new supply is in, be sure to carefully touch that connection and see if it's heating up. I will pass the connection if it's barely perceptibly warm but fail it if I feel it's warm or hot. Heat there means a poor connection and a fire hazard.
My computer started crashing while playing games, I noticed burn marks on the cables that connected the power supply to the GPU, I took it out and examined the GPU slot and 1 of the pin ports has some melted plastic in it, I've ordered a new PSu but does this mean my graphics card is finished? What should I do? I'm a student so buying a new graphics card is gonna cost a lot for me