Endlessly pester about emblem (5)
I believe the answer is:
badge
'emblem' is the definition.
(badge is a kind of emblem)
'endlessly pester' is the wordplay.
'endlessly' means to remove the last letter.
'pester' becomes 'badger' (synonyms).
'badger' with its last letter removed is 'BADGE'.
'about' is the link.
(Other definitions for badge that I've seen before include "Emblem showing membership etc" , "Label with name or status" , "evidence of membership" , "Symbol worn, to indicate membership perhaps" , "Symbol worn for identification" .)