Teacher at university that’s not quite finished (3)
I believe the answer is:
don
'teacher at university' is the definition.
'not quite finished' is the wordplay.
'not quite' means to remove the last letter.
'finished' becomes 'done' ('done' can be similar in meaning to 'finished').
'done' with its last letter removed is 'DON'.
'that's' acts as a link.
(Other definitions for don that I've seen before include "Member of university" , "put on > one of several rivers" , "University teacher" , "Mafia boss" , "Professor" .)