After half-term, extremely short exam (4)
I believe the answer is:
test
'exam' is the definition.
('test' can be a synonym of 'exam')
'after half-term extremely short' is the wordplay.
'after' is a charade indicator (letters next to each other).
'half-term' becomes 'te'.
'extremely' means to remove the middle letters.
'short' with its centre removed is 'st'.
'te' next to 'st' is 'TEST'.
(Other definitions for test that I've seen before include "water flowing through Hampshire" , "Critical examination or trial of qualities or abilities" , "Cricket match; try" , "Trial; cricket match" , "Evaluation" .)