Annoying person; destructive insect (4)
I believe the answer is:
pest
This clue is a double definition.
'annoying person' is the first definition.
(pest can mean an annoying person)
'destructive insect' is the second definition.
(pest can mean insect that damages crops)
(Other definitions for pest that I've seen before include "Annoying creature" , "A nuisance - step on it" , "Destructive insect or nuisance" , "Destructive creature" , "Annoying person" .)