Peer, prime minister or civic dignitary (4,5)
I believe the answer is:
lord mayor
'civic dignitary' is the definition.
(I've seen this before)
'peer prime minister or' is the wordplay.
'peer' becomes 'lord' (I've seen this before).
'prime minister' becomes 'may' (I've seen this in other clues).
'lord'+'may'+'or'='LORD MAYOR'
(Other definitions for lord mayor that I've seen before include "magistrate" , "city position" , "VIP in London, perhaps" , "Showman" , "Municipal official" .)