Region getting better score, with time (6)
I believe the answer is:
sector
'region' is the definition.
(sector is a kind of region)
'better score with time' is the wordplay.
'better' is an anagram indicator (with an improved letter-order).
'with' indicates putting letters inside.
'time' becomes 't'.
'score' with letters rearranged gives 'secor'.
'secor' placed around 't' is 'SECTOR'.
'getting' is the link.
(Other definitions for sector that I've seen before include "Area of circle between two radii" , "battle zone" , "Distinct area or portion" , "Division or part" , "region" .)