Graduate against some breakfast (5)
I believe the answer is:
bacon
'some breakfast' is the definition.
'graduate against' is the wordplay.
'graduate' becomes 'ba' (someone with a Bachelor of Arts degree).
'against' becomes 'con' (as in pros and cons, for and against).
'ba'+'con'='BACON'
(Other definitions for bacon that I've seen before include "Salted and smoked pig meat" , "gammon" , "What the breadwinner brings home?" , "Cut of pork" , "Meat - essayist - painter" .)