Ape behind, bird overhead (7)
I believe the answer is:
emulate
'ape' is the definition.
(ape can mean to imitate or emulate)
'behind bird overhead' is the wordplay.
'behind' becomes 'late' (as in behind schedule).
'bird' becomes 'emu' (emu is a type of bird).
'overhead' means one lot of letters go next to another (in a down clue, some letters go over others).
'late' after 'emu' is 'EMULATE'.
(Other definitions for emulate that I've seen before include "Try to do as well as, copy" , "Try to equal or surpass (someone)" , "Imitate - rival" , "Copy - rival" , "Try to surpass what one admires" .)