Chase after fellow to supply bark (4)
I believe the answer is:
woof
'bark' is the definition.
(I know that bark can be written as woof)
'chase after fellow' is the wordplay.
'chase' becomes 'woo' (to woo is to pursue romantically).
'after' says to put letters next to each other.
'fellow' becomes 'f' (abbreviation - of a society etc.).
'woo'+'f'='WOOF'
'to supply' acts as a link.
(Other definitions for woof that I've seen before include "from the setter?" , "Toto's line?" , "Sound of bark - weft" , "As dog goes" , "Sound of dog's bark" .)