Finger damaged by this edge (6)
I believe the answer is:
fringe
'this edge' is the definition.
(fringe is a kind of edge)
'finger damaged' is the wordplay.
'damaged' indicates an anagram.
'finger' anagrammed gives 'FRINGE'.
'by' is the link.
(Other definitions for fringe that I've seen before include "Rim" , "Border of threads or hair" , "Perimeter" , "lunatics, perhaps" , "Ornamental border of threads" .)