Score that's zero for golfer (7)
I believe the answer is:
scratch
I believe this is a double definition.
'score' is the first definition.
(both can mean to mark or etch)
'zero for golfer' is the second definition.
(scratch handicap is a zero handicap in golf)
'that's' is the link.
(Other definitions for scratch that I've seen before include "Write hurriedly" , "Back out of race" , "Superficial damage; withdraw" , "handle superficial irritation" , "having superficial injury" .)