Hicarer 50 Pack R-Type Cable Clip Wire Clamp, Nylon Screw Mounting Cord Fastener Clips with 50 Pack Screws for Wire Management (White,1/4 Inch)

Get some clips like this: https://www.amazon.com/Hicarer-R-Type-Mounting-Fastener-Management/dp/B07DFD5X14 They can easily screw into the underside of the desk to keep the cable in place, and should work for the cable connector as well. Then get a headphone cable extension cable, and run that from your pc to the underside front of the desk. Now you can plug your headphones into the new port underneath your desk, and unplug them and tidy them away when you need. This is exactly what I do and it works great, though I drilled a hole and actually recessed the extension cable connector into the front edge of the desk.