Yes, I agree with Leo, it would be possible.
You could also use some epoxy tape type stuff, available from motor factors for sealing exhaust pipes... you'd use this first on the underside of the sink, let it set, and then fill in from above using an epoxy paste.
The exhaust stuff is probably unneccessary.. but you do need something on the back.. so duct tape would work too.
You can get white two part wood filler from hardware or paint shops.. (Woodworkers if in Dublin). Woodworkers sell a little tin of white 2 part woodfillers for about 7.50. (Very similar to car body fillers and fully waterproof when set)
The repair wil be noticeable but it need not look terrible.. it all depends on how well you carry out the repair.
If the backside of the sink is in-accessible then the repair would be much harder, and it may not work.