Monarch embarrassed having sinned (5)
I believe the answer is:
erred
'sinned' is the definition.
(I've seen this in another clue)
'monarch embarrassed' is the wordplay.
'monarch' becomes 'ER' (abbreviation for Elizabeth Regina, British Queen).
'embarrassed' becomes 'red' (someone may blush or go red when embarrassed).
'er'+'red'='ERRED'
'having' is the link.
(Other definitions for erred that I've seen before include "Committed a sin" , "was inaccurate" , "Went wrong" , "Gone astray!" , "Sinned" .)