Should I run cable through pipe?
Better plan: cut out the damaged drywall and install a low-voltage bracket and coax barrel wallplate

Some cheap-arse folks do what you see here -- simply punch a hole in the drywall to run low-voltage cabling through.  While this sort-of-works, it leads to ugly results like the ones you see here.  It's much better to do the neat and tidy thing and install a low-voltage old work bracket and coax barrel wallplate instead.

 

Installation is as follows:


Cut out the damaged drywall to match the cutting template for the old work bracket you're using
Install the old-work bracket using the clamps that come with it -- you'll have to run the coax through it first to do this in your case
Undo the existing barrel connection between the two lengths of coax
Screw the coax from the wall onto the back of the barrel wallplate
Attach the wallplate to the bracket
Attach the removed coax cable to the front of the wallplate