After a bad start, is half-hoping to see primate, perhaps (6)
I believe the answer is:
bishop
'primate perhaps' is the definition.
(I know that primate (bishop) is a type of bishop)
'after a bad start is half-hoping' is the wordplay.
'after' says to put letters next to each other.
'start' indicates taking the first letters.
'half' suggests halving the letters.
The initial letter of 'bad' is 'b'.
'hoping' cut in half is 'hop'.
'b'+'is' is 'bis'.
'bis'+'hop'='BISHOP'
'to see' acts as a link.
(Other definitions for bishop that I've seen before include "Church leader" , "He wears the mitre and carries the crozier" , "Chess piece and senior religious person" , "Chess piece and one of the clergy" , "see supervisor" .)