Peter managed with acceptable name (6)
I believe the answer is:
repute
'name' is the definition.
(name can mean someone's reputation)
'peter managed with acceptable' is the wordplay.
'managed with' is an anagram indicator.
'acceptable' becomes 'u' (as in U and non-U speech).
'peter' with its letters mixed up with 'u' makes 'REPUTE'.
(Other definitions for repute that I've seen before include "Fame, renown" , "General view or opinion of an individual, may be ill" , "Known character, standing" , "High estimation; fame" , "Public estimation" .)