PC Master Race

Looks like a B550 series board which means those two headers on the motherboard would be chassis fan 1 and aio pump. If you look closely on the motherboard it should be labeled near the header (in the pic you can just barely read "ch" on the left side and "aio pump" on the right.) The cable looks like a molex to 3 or 4 pin power adapter which would've been used with an older chassis fan with a molex connector. If everything is already hooked up you can remove that adapter.

Edit: you can also look up your specific motherboard model and download the user manual which will give you a detailed layout of all of the headers on the board.

It looks like a molex cable to me which is just power. It will probably be daisy chained together with 2 other of the same connector that then goes to your PSU. The other parts could be connected to something. If possible I would switch it out with Sata power anyway. I don't trust molex.

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.

Does it attach to the motherboard in one end? As in is one end still attached? If so is there any labels / text there? It looks like it could be a fan cable and if it is the end connected to the motherboard should have a label.

If not that I'd need more pictures, especially of the loose end and the connection point in the connected end.

I don't think this is related to your headphones not working though.

It’s for a fan.

That fan can be powered either with the Molex connector (big chunky loose/floating bit) or a PWM fan connector (smaller one plugged in to the motherboard).

You wouldn’t plug in both of them at the same time.

Since there’s already a connection directly to the mainboard, the bigger connector was already disconnected and probably tucked under the IO shield.l that you taped it to.

If you want, you could also route the cable over to the back of the chassis and plug it in to the power supply’s Molex cable. Instead of having its speed regulated by the motherboard, it would just run at 100% all the time.