Flier bringing dire distress to an egghead (5)
I believe the answer is:
eider
'flier' is the definition.
(eider is a kind of birdanimal)
'dire distress to an egghead' is the wordplay.
'distress' indicates anagramming the letters.
'to' is a charade indicator (letters next to each other) (I've seen this in other clues).
'an egghead' becomes 'e' (head letter of 'egg').
'dire' anagrammed gives 'ider'.
'ider' after 'e' is 'EIDER'.
'bringing' acts as a link.
(Other definitions for eider that I've seen before include "Sea-duck" , "Downy sea duck" , "Sea bird" , "Duck with soft feathers" , "Duck known for down" .)