Bête noire, pet hate (7)
I believe the answer is:
bugbear
I believe this clue is a double definition.
'bete noire' is the first definition.
(I've seen this before)
'pet hate' is the second definition.
(I've seen this before)
(Other definitions for bugbear that I've seen before include "Cause of obsessive fear" , "A source of annoyance" , "constant worry" , "Particular irritation" , "Nuisance" .)