Neighbour, source of annoyance with objection (4)
I believe the answer is:
abut
'neighbour' is the definition.
(I've seen this before)
'source of annoyance with objection' is the wordplay.
'source of' indicates taking the first letters.
'with' is a charade indicator (letters next to each other).
'objection' becomes 'but'.
The initial letter of 'annoyance' is 'a'.
'a'+'but'='ABUT'
(Other definitions for abut that I've seen before include "Lie adjacent to" , "Adjoin, be contiguous to" , "Touch upon" , "Adjoin, border" , "End or lean against, say a building" .)