A keenness to get the advantage (4)
I believe the answer is:
edge
I believe this clue is a double definition.
'a keenness' is the first definition.
(I've seen this in another clue)
'the advantage' is the second definition.
(having the edge is having an advantage)
'to get' acts as a link.
(Other definitions for edge that I've seen before include "Border; intense quality" , "Some bordering" , "Sidle, lip" , "Outer limit" , "Brink or border" .)