English royal embarrassed, having made a mistake (5)
I believe the answer is:
erred
'made a mistake' is the definition.
('err' can be a synonym of 'make a mistake')
'english royal embarrassed' is the wordplay.
'english' becomes 'E' (abbreviation).
'royal' becomes 'r' (common abbreviation e.g. RM - Royal Marine).
'embarrassed' becomes 'red' (someone may blush or go red when embarrassed).
'e'+'r'+'red'='ERRED'
'having' is the link.
(Other definitions for erred that I've seen before include "slipped up" , "Was mistaken" , "Committed a sin" , "Made mistake, was incorrect" , "Went wrong" .)