How can I stop an air leak behind a cabinet?
Possibly caused by some problem with the dish-washer?

It seems to be structurally sound enough. (maybe 40% strength, but "enough")

Fill the void with expanding foam and place a wooden panel in the lower part of the back (do no get in the way of the drawer) to keep the foam in while it cures, the foam will glue it in place forming a new back for that cupboard.