Annoyed by Dire Straits, unfortunately twice missing start of song (9)
I believe the answer is:
irritated
'annoyed' is the definition.
('irritate' can be a synonym of 'annoy')
I can't explain the rest of the clue.
(Other definitions for irritated that I've seen before include "Vexed" , "Stimulated" , "Irked" , "put out" , "Ratty" .)