Control board powered up with reverse polarity. power transistor blown. Is this repairable?
I'm guessing the black blobs are tar formed from the resin that was overheated.

one of them is covering the label for this part. it looks like it might start with U.

That's a good thing. because it means this burnt thing is probably a voltage regulator, and if that's the case it probably took all the energy from the wall wart, and is the only damaged part on the board.

the PCB is blackened and that black is carbon, so it probably now conducts electricity in places where it should not.  but as the voltage is low and the damage localised that's probably not going to significantly impede function.

but now the problem is choosing the right replacement. the marking on it 
seems to say L78xx but the last two digits are unreadable in the photo. however knowing the number is vital choosing the right part. 

It's probably not L7805 as there's already one of those U6 and it would be unusual to use two the same.9V is not enough for L7808 so the part is probably L7806, but that's just a guess.