Rome’s increasingly spoilt, at first, for choice of popes? (4)
I believe the answer is:
pius
'choice of popes?' is the definition.
(papal name chosen by several popes)
'rome's increasingly spoilt at first' is the wordplay.
'rome's increasingly' becomes 'piu' ('more' in Italian eg in Rome).
'at first' says to take the initial letters.
The initial letter of 'spoilt' is 's'.
'piu'+'s'='PIUS'
'for' is the link.
(Other definitions for pius that I've seen before include "Name of twelve Popes" , "old pope" , "WW two Pope (XII)" , "religious leader" .)