Fairy queen embraced by Greek character (4)
I believe the answer is:
peri
'fairy' is the definition.
(Persian mythological being)
'queen embraced by greek character' is the wordplay.
'queen' becomes 'er' (abbreviation for Elizabeth Regina).
'embraced by' is an insertion indicator.
'greek character' becomes 'pi' (letter from the Greek alphabet).
'er' put into 'pi' is 'PERI'.
(Other definitions for peri that I've seen before include "Supernatural being in Persian folklore" , "Mythical being" , "Spirit of folklore" , "A vision of loveliness" , "Fairy (Pers. mythology)" .)