The longing teachers have at heart (4)
I believe the answer is:
ache
'the longing' is the definition.
(longing can mean aching or pining for something)
'teachers have at heart' is the wordplay.
'at heart' indicates a hidden word (letters hidden in the heart of the clue).
'ACHE' is hidden in the letters of 'teachers have'.
(Other definitions for ache that I've seen before include "Dull throbbing pain" , "Similar" , "Long ? pain" , "Persistent dull pain" , "Hanker" .)