Annoyed Irish king – extremely exasperated (5)
I believe the answer is:
irked
'annoyed' is the definition.
(I've seen this before)
'irish king extremely exasperated' is the wordplay.
'irish' becomes 'IR' (abbreviation).
'king' becomes 'K' (abbreviation for king in chess).
'extremely' suggests removing the centre (only the extremes of the word are used).
'exasperated' with its middle taken out is 'ed'.
'ir'+'k'+'ed'='IRKED'
(Other definitions for irked that I've seen before include "Galled" , "Vexed, annoyed" , "Irritated, vexed" , "Wound" , "Exasperated" .)