Elizabethan playwright’s horse crossing down (7)
I believe the answer is:
marlowe
'elizabethan playwright's' is the definition.
(Christopher Marlowe)
'horse crossing down' is the wordplay.
'horse' becomes 'mare' (mare is a kind of horse).
'crossing' means one lot of letters goes inside another (to cross can mean to go over or around).
'down' becomes 'low' (similar in meaning).
'mare' placed around 'low' is 'MARLOWE'.
(Other definitions for marlowe that I've seen before include "Christopher -, playwright (Tamburlaine)" , "Christopher -, sixteenth English dramatist" , "Author" , "writer" , "Christopher ..., playwright of 'Dr Faustus'" .)