If the voice coil is burnt out, then it sounds like a snare drum when playing and has relatively little power.
It sounds more like the braided flex wires that connect to the VC from the terminals might have snapped - if so, a quick soldering will fix it.
Check the resistance over the terminals on the speaker - it should give some insight into what's going on.
Subs can be repaired, but to be honest, its only usually worth it if is actually valuable in the first place > £1-200