Crossword Genius

A flier, but the fathead doesn't get far! (5)

Ross

I believe the answer is:

finch

I'm a little stuck... Click here to teach me more about this clue!

'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" .)

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