Diffident about island, and right annoyed (6)
I believe the answer is:
shirty
'annoyed' is the definition.
(I've seen this before)
'diffident about island and right' is the wordplay.
'diffident' becomes 'shy' (synonyms).
'about' means one lot of letters goes inside another.
'island' becomes 'i' (geographical abbreviation).
'and' is a charade indicator (letters next to each other).
'right' becomes 'rt' (abbreviation as in Rt Hon- the Right Honourable).
'i'+'rt'='irt'
'shy' enclosing 'irt' is 'SHIRTY'.
(Other definitions for shirty that I've seen before include "Irritable" , "Tetchy" , "Ill-tempered or annoyed with a garment" , "Bad-tempered (like a garment?)" , "Ill-tempered (informal)" .)