Annoy Irish coming over in improper attire (8)
I believe the answer is:
irritate
'annoy' is the definition.
('irritate' can be a synonym of 'annoy')
'irish coming over in improper attire' is the wordplay.
'irish' becomes 'ir' (abbreviation).
'coming over' is a reversal indicator.
'in' is an insertion indicator.
'improper' is an anagram indicator.
'ir' written backwards gives 'ri'.
'attire' is an anagram of 'irtate'.
'ri' going within 'irtate' is 'IRRITATE'.
(Other definitions for irritate that I've seen before include "Rattier, I may be annoying" , "Annoy, rankle with" , "Nettle and annoy" , "Make angry or fretful" , "Needle" .)