Bird's heading off — a very short distance! (4)
I believe the answer is:
inch
'a very short distance' is the definition.
(I've seen this before)
'bird's heading off' is the wordplay.
'bird' becomes 'finch' (finch is a kind of bird).
'heading off' means to remove the first letter.
'finch' with its first letter taken off is 'INCH'.
(Other definitions for inch that I've seen before include "Imperial measure" , "Move by small amounts" , "Edge slowly" , "Two point five four cm" , "One 63,360th of a mile" .)