Bird heading off a small distance (4)
I believe the answer is:
inch
'a small distance' is the definition.
(I have seen 'Short distance' mean 'inch' so perhaps 'distance' could also mean 'inch'; 'small' asks for an abbreviation)
'bird 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 "Edge towards" , "Move in tiny increments" , "Subdivision of 14 down" , "Small Scottish island" , "Twenty-five point four millimetres" .)