It looks like a fan header with a molex split. But more pictures would be needed to confirm 100% (A picture of where the cable connects to the motherboard in particular)
I assume it is a fan with both a 4pin header and molex like https://ae01.alicdn.com/kf/HTB1bjiuX5frK1RjSspbq6A4pFXaS/120mm-LED-Ultra-Silent-Computer-PC-Case-Fan-15-LEDs-12V-With-Rubber-Quiet-Molex-Connector.jpg You would only connect one of the ends and the other end would be left disconnected.
My guess is that the molex end was never connected to anything and was simply tucked under the IO shield to the left or the VRM heat sink above. It was probably jostled loose during transport. You can take a look around for anything with the same size as the loose connector to see if something was plugged in there. The connector is shaped like the letter “D” where one side is flat and the other has slightly rounded corners so you can only plug it in one way. This cable is for power only so it would probably be something like a fan or a light.
If I am correct then the cable can be left unplugged and you can tuck it somewhere to hide it or just leave it taped. It would also be unrelated to your headphone issue.
I started with a $15 double edge safety razor that I used for years. I bought the $70 Henson razor because I wanted a better shave, but the results are about the same as the $15 razor I started with. The Henson razor glides better so I can get the same quality shave with less time. I can also shave twice (one across the grain once against) if I want a better shave, but it still isn’t the smooth shave I was hoping for. The henson razor came with 100 blades, but you can also get 100 blades for $10 on amazon.