Insect on animal's a source of annoyance (7)
I believe the answer is:
bugbear
'a source of annoyance' is the definition.
(I've seen this before)
'insect on animal's' is the wordplay.
'insect' becomes 'bug' (bug is a kind of insect).
'on' means one lot of letters go next to another.
'animal' becomes 'bear' (bear is a kind of animal).
'bug'+'bear'='BUGBEAR'
(Other definitions for bugbear that I've seen before include "Nuisance" , "Bete noire" , "One's special fear" , "It causes obsessive fear" , "Object of obsessive, foolish fear" .)