Tape all the way around the sink where you want the caulk to stop. Tape the top edge around the linoleum and trim it around the rough area with a razor blade on the left side. Throw a fat bead of kitchen sink caulk in there and lay it off with a wet finger if its water soluable. Wipe the excess off on a paper towel. Wait till the caulk sets up a little and unmask the tape or fill in any low spots, lay it off with your finger again then wait and pull tape. Cut that caulk tube wide open.
Not sure if this is an option but if you totally drop the sink you can remove all the old crap and then put new sealer and remount the sink.