Pester endlessly for distinguishing emblem (5)
I believe the answer is:
badge
'distinguishing emblem' is the definition.
(a badge is an emblem or sign)
'pester endlessly' is the wordplay.
'pester' becomes 'badger' (synonyms).
'endlessly' means to remove the last letter.
'badger' with its last letter taken off is 'BADGE'.
'for' acts as a link.
(Other definitions for badge that I've seen before include "Symbol worn, to indicate membership perhaps" , "Identifying sign" , "Thing worn to show, say, membership of organisation" , "Emblem to denote membership" , "Emblem, to symbolise membership perhaps" .)