Annoyed Irish king – extremely exasperated (5)
I believe the answer is:
'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' means to remove the middle letters (only the extremes of the word are used).
'exasperated' with its middle taken out is 'ed'.
(Other definitions for irked that I've seen before include "Galled" , "Vexed, annoyed" , "Irritated, vexed" , "Wound" , "Exasperated" .)