I'm not certain, but I think the antenna cable is coaxial (the inner plastic is probably just for insulation). If it's coax, you may have to just replace it
. I've never had good luck fixing that kind of wire.
If your amp supports high level inputs (most modern ones do), you should just tap into the factory sub's wiring. That way you can control the level (a bit). Also, you can simply split and splice that wire into the L and R inputs to your stereo amp (what I did).
If you are seeing some issues with louder bass during acceleration, you either have a damaged alternator/battery, or you are drawing too much power from ur electrical system and you may be damaging it. Consider a higher power alternator/battery and a high capacity capacitor.