Pursue mate - not in Kent area (5)
I believe the answer is:
chase
'pursue' is the definition.
(chasing is a kind of pursuing)
'mate not in kent area' is the wordplay.
'mate' becomes 'china' (cockney rhyming slang, mate is china plate).
'not' is a deletion indicator.
'kent area' becomes 'se'.
'china' with 'in' taken out is 'cha'.
'cha'+'se'='CHASE'
(Other definitions for chase that I've seen before include "Engrave - hunt - pursue" , "Follow" , "Pursuit" , "Hurry" , "Hound" .)