A flier, but the fathead doesn't get far! (5)
I believe the answer is:
finch
'a flier' is the definition.
(finch is a kind of birdanimal)
'the fathead doesn't get far' is the wordplay.
I cannot really see how this works, but
'fathead' could be 'f' (head letter of 'fat') and 'f' is found within the answer.
The remaining letters 'inch' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
'but' is the link.
Can you help me to learn more?
(Other definitions for finch that I've seen before include "Singer" , "Seed-eating songbird" , "Type of bird" , "Bird such as canary" , "Songbird such as a canary" .)