The black "plastic mount thing" carrys the water from the overflow hole in the bowl to the bowl tailpiece drain pipe. If it seems like its just a pin hole in that blue sealant, You should be fine with some clear silicone caulk. Give it 24 hours to dry, even if it says it drys in less.
I just bought my first house and it needs a lot of updates. For the living room, dining room, and 2 bed rooms, I plan to paint the walls, sand and restore the hardwood floors, have new windows installed.
My question is what would be the best order to do those 3 things?