Bird of prey old playwright’s written in Greek (7)
I believe the answer is:
goshawk
'bird of prey' is the definition.
(goshawk is a kind of bird of prey)
'old playwright's written in greek' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'playwright' becomes 'shaw' (George Bernard Shaw).
'written in' indicates putting letters inside.
'greek' becomes 'gk'.
'o'+'shaw'='oshaw'
'oshaw' put into 'gk' is 'GOSHAWK'.
(Other definitions for goshawk that I've seen before include "Short-winged bird of prey" , "Bird used in falconry" .)