Stay to consume university leftovers (7)
I believe the answer is:
residue
'leftovers' is the definition.
(thesaurus)
'stay to consume university' is the wordplay.
'stay' becomes 'reside' (I've seen this in other clues).
'to consume' indicates putting letters inside.
'university' becomes 'u'.
'reside' going around 'u' is 'RESIDUE'.
(Other definitions for residue that I've seen before include "Surplus" , "Matter left behind after other parts have been taken away" , "Remaining portion" , "rest" , "Small amount remaining" .)