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 within 'teachers have'.
(Other definitions for ache that I've seen before include "Dull throbbing pain" , "Similar" , "Long ? pain" , "Persistent dull pain" , "Hanker" .)