Looks like it is deep enough that it went to bare wood in places, so you won't be getting it out.
If I were to attempt this, I would start by using a very sharp knife to remove all of the loose, jagged parts. Extreme care is called for, as is a good magnifying glass and light.
If the finish appears to be lifted in places, you might fill with thin superglue--be very careful about this application as it will go everywhere if too much is applied. I like to use a toothpick or popsicle stick to apply a drop at a time.
Next, carefully fill the gouges with either thick superglue or epoxy to build up slightly higher than the original surface. Clean the area with isopropyl alcohol before application(s)--it may well take multiple applications.
Finally, carefully sand the built-up area to flush with the surface. Depending on how high it is built up, you shouldn't need anything coarser than 600grit sandpaper--always start finer than you think you will need. There are great videos on youtube of guys doing this on guitars.
Finally, you will need to buff the finish--at this point you will be working a larger area than the damage, possibly the entire forearm or cue. This can be done by hand with quality automotive polishes (i.e. Maguire's, 3M, or Griot's). You will need a few different products starting with a medium or coarse cut compound and working to fine cut cleaner and glaze coat.
You may still see evidence of the damage, especially with a problem as large as yours, but you won't feel it and should have to look for it.
This is really a pretty simple procedure, but the final result will depend on your skill and the amount of care/time invested.
If you don't have any of the materials listed above, you will likely have $60-$100 in the repair. YMMV if it is worth that over sending it out for a refinish, but keep that in mind if the refinish quote seems high.
