Our cousins a nuisance, almost (4)
I believe the answer is:
apes
'our cousins' is the definition.
(evolutionary 'cousins' of humanity)
'a nuisance almost' is the wordplay.
'nuisance' becomes 'pest' (I've seen this before).
'almost' means to remove the last letter (most of the word but not all of it).
'pest' with its final letter removed is 'pes'.
'a'+'pes'='APES'
(Other definitions for apes that I've seen before include "Zoo animals" , "Parrots" , "Peas (anag)" , "Imitates like primates" , "Species of primate" .)