How can I repair a hole in the underside of a granite counter?
I am guessing that your sink weight is causing the pressure, making the cracks worse.  If I am right on this then the first thing you need to do is build a simple way to support the sink using wood or whatever - I have 2x2s that are screwed to side cabinet and pushed up to support it.  So that will cost you $3.  If side walls are not close enough you have to build a box.  

Now to fix the cracks you will need a knife grade epoxy.  I have used Tenax brand in the past to "glue" granite corner shelves in the shower.  Pretty easy to use.  Epoxy will be about $20-$30.

And for some reason (conspiracy) the big boxes don't carry any knife grade epoxy.  You will have to hit the interwebs for this. Just search for knife grade granite epoxy if you want to read reviews or whatever.  Basically you will get the activator and the putty.  You put a little activator in the putty and wipe it on the granite.  If you are doing the top then you might need to by color matching kit (which is more expensive than epoxy).  Basically it is hard in 30 mins and can be used the next day.  Once epoxy is on it is permanent and harder than the granite.  If you arent worried about looks your sink area could be cured of the crack as long as you get the epoxy in the right places.