Obnoxious; highly offensive (4)
I believe the answer is:
vile
I believe this clue is a double definition.
'obnoxious' is the first definition.
(vile means obnoxious or unpleasant)
'highly offensive' is the second definition.
(I've seen this in another clue)
(Other definitions for vile that I've seen before include "Book by Evelyn Waugh" , "Live (anag.)" , "Evil - evil (anag)" , "Evil, foul" , "Extremely unpleasant" .)