A fathead won’t get far as a flier! (5)
I believe the answer is:
finch
'flier' is the definition.
(finch is a kind of birdanimal)
'a fathead won't' is the wordplay.
'a fathead' becomes 'f' (head letter of 'fat').
'won't' becomes 'inch' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'f'+'inch'='FINCH'
'get far as a' acts as a link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for finch that I've seen before include "Peter - - , film actor; bird" , "Type of bird" , "Songbird such as a canary" , "Singer" , "Seed-eating songbird" .)