A bird going after bee, endlessly (and fly) (4,2)
I believe the answer is:
beat it
'fly' is the definition.
(I've seen this before)
'a bird going after bee endlessly' is the wordplay.
'bird' becomes 'tit' (tit is a kind of bird).
'going after' says to put letters next to each other.
'endlessly' means to remove the last letter.
'bee' with its final letter taken away is 'be'.
'a'+'tit'='atit'
'atit' after 'be' is 'BEAT IT'.
'and' is the link.
(Other definitions for beat it that I've seen before include "Scarper" , "Sling your hook!" , "Slang order to go away, get lost" , "try out drum kit?" , "leave" .)