Is my supervisor allowed to push a crate with a monkey on top?
YES you can push a crate with either Monkey on top BUT ONLY if you can afford it (as suggested on the Super Speed power card)...
Moving Supervisor = 1AP
Moving Crate = 1AP
Fighting with the Monkey = 3AP
TOTAL to Push Crate with Monkey on top = 5AP
Therefore in a normal turn where you only have 4AP to spend this move is not possible, but with the Super Speed +1AP power card you can