That is the 4 pin connector for your CPU, which supplies the CPU power. It is most definitely burnt. Your CPU may also not work now, as this may have damaged the CPU.
So replacing the cable would be a good idea i guess.
Motherboard: Gigabyte M61PME-S2P PSU: Codegen Superpower 300x (Manufacturer Codegen, model 300x. There's not much on the internet about this one) I'm using this PSU for over 6 years. I connected it and the computer works fine. Just wondering if this can become a problem.
I would consider replacing your power supply. I think it is well overdue for a new psu.