Church dignitary (7)
(Other definitions for primate that I've seen before include "Archbishop - monkey, perhaps" , "bishop, perhaps" , "Metropolitan" , "Man in the mitre" , "Higher mammal" .)
'church dignitary' is the definition.
(I've seen this before)
This is all the clue.
(Other definitions for prelate that I've seen before include "Senior member of the clergy" , "Bishop or other ecclesiastical dignitary" , "High-ranking member of clergy, say bishop" , "e.g. Bishop" , "High-ranking clergyman" .)
(Other definitions for provost that I've seen before include "College head - Scots mayor" , "College head; minister" , "The head of certain colleges like Trinity, Dublin" , "(In Scotland) chief district councillor - college head" , "A college head; a mil. policeman" .)
'church dignitary' is the definition.
'dean' can be an answer for 'dignitary' (I have seen 'Cathedral dignitary ' mean 'dean' so perhaps 'dignitary' could also mean 'dean'). I also know that 'church' relates to this answer.
This is the entire clue.
(Other definitions for dean that I've seen before include "Head of e.g. faculty" , "Cathedral chapter head" , "Glos. forest; Dixie ---, Everton hero" , "Cathedral dignitary; head of university department" , "University officer" .)