Stay up studying (4)
I believe the answer is:
cram
'stay up studying' is the definition.
(I've seen this in another clue)
This is all the clue.
(Other definitions for cram that I've seen before include "Stuff (facts for exam)" , "Stuff full" , "Quickly learn - stuff" , "prepare hastily for the paper" , "Pack in or study hard" .)