Sound in one headphone keeps going off and on, moving the wire around can fix it temporarily but is very annoying

You have a break in the cable. You don't say in what area it's in, so I will have to give you 3 ways of clearing the fault.
1: Break in, near or around the plug. The best solution is to cut off the plug making certain the break area goes with it. You can then fit a new 3.5mm plug. If you need advice on how to do this let me know.
2. Break halfway to 2-thirds along the cable. Again cut the cable to a point clear of the break, again fit a new plug. To extend it you can purchase an extension headphone cable from any hi-fi shop.
3. Break to close to headset. In which case you will need to replace the entire lead. To do this you need to open up the headphones. Access is obtained by removing the pad(s) and any foam. You might need to force the foam off. You can glue it back on afterwards. You should then see screws to open it.
Fitting new plugs or attaching the wire inside will require the use of a small soldering iron.