Egghead goes after set of parts to make flier (4)
I believe the answer is:
kite
'flier' is the definition.
(kite is a kind of birdanimal)
'egghead goes after set of parts' is the wordplay.
'egghead' becomes 'e' (head letter of 'egg').
'goes after' is a charade indicator (letters next to each other).
'set of parts' becomes 'kit' (I've seen this in another clue).
'e' put after 'kit' is 'KITE'.
'to make' acts as a link.
(Other definitions for kite that I've seen before include "Flying toy; a quality mark" , "Small hawk" , "Toy; aircraft (dated)" , "Flying toy or kind of hawk" , "A flying toy or sea bird" .)