Education after half-term's suffering (8)
I believe the answer is:
teaching
'education' is the definition.
(teaching is a kind of education)
'after half-term's suffering' is the wordplay.
'after' says to put letters next to each other.
'half-term' becomes 'te'.
'suffering' becomes 'aching' ('ache' can be a synonym of 'suffer').
'te' put next to 'aching' is 'TEACHING'.
(Other definitions for teaching that I've seen before include "Educating" , "Imparting skills or knowledge" , "exercise ..." , "Training" , "Instruction" .)