Hmmmm. At least from my experience it appears that this counter top may have been cut out for the sink improperly. It does not need to have that much remaining surface of counter top at back of sink. It should be cut out with only about 3/8th in lip on each side. Only about 3/4" smaller than the sink in width and depth. Having not been done on back side it required the installer to drill these holes to run supply lines, making it harder to install supply lines and much more prone to water damage. If done right this wouldnt have been an issue. But have faith there is hope.
Fix what has been done
It appears the damaged area of the counter top doesn't need to be and shouldn't be there. In fact almost all stainless sinks require this part of the counter to not be there as for this is where the track is for the back sink clips. Do this:
1. place the sink in the cutout
2. trace a line around the back side
3. measure in 3/8ths of an inch and draw another line (use actual corners of sink to trace pattern of corners)
4. cut out remaining material which appears to be damaged
Counter tops should be cut from the bottom for ease and to avoid chipping the laminate on the saws up-stroke. To avoid having to work under sink or removing counter top to cut, you may want to take a straight edge and a razor and go over your cut line numerous times before cutting, this will avoid chipping. It looks like the cut would be pretty close to the back splash, so you may wanna try a jig saw. If you're a diyer, whatever you use, the cut doesnt necessarily have to look decent as long as you stay near your lines as the sink will cover imperfections.