Clytemnestra's child chosen by hawk-headed god (7)
I believe the answer is:
electra
'clytemnestra's child' is the definition.
(Greek mythology)
'chosen by hawk-headed god' is the wordplay.
'chosen' becomes 'elect' ('elect' as an adjective can mean chosen).
'by' says to put letters next to each other.
'hawk-headed god' becomes 'ra' (Egyptian god).
'elect'+'ra'='ELECTRA'
(Other definitions for electra that I've seen before include "female in tragedy" , "Complex parallel to Oedipus" , "Female avenger" , "Daughter of Agamemnon and Clytemnestra (who has a complex?)" , "Greek heroine (with a complex!)" .)