Neighbour having an objection (4)
I believe the answer is:
abut
'neighbour' is the definition.
(I've seen this before)
'an objection' is the wordplay.
'an' becomes 'a'.
'objection' becomes 'but'.
'a'+'but'='ABUT'
'having' acts as a link.
(Other definitions for abut that I've seen before include "Impinge" , "Adjoin at one end" , "Adjoin, be contiguous to" , "Touch (side of next building)" , "Lean against the odd tuba" .)