Teacher’s mild rebuke over lacking heart (5)
I believe the answer is:
tutor
'teacher's' is the definition.
(tutor is a kind of teacher)
'mild rebuke over lacking heart' is the wordplay.
'mild rebuke' becomes 'tut'.
'lacking heart' says to hollow out the word (remove centre letters).
'over' with its centre removed is 'or'.
'tut'+'or'='TUTOR'
(Other definitions for tutor that I've seen before include "Personal coach" , "trainer" , "Trout (anag)" , "Give individual instruction" , "Private instructor" .)