Run finished in outskirts of Gijon (6)
I believe the answer is:
govern
'run' is the definition.
(both can mean to rule over)
'finished in outskirts of gijon' is the wordplay.
'finished' becomes 'over' (over can mean finished or ended).
'in' means one lot of letters goes inside another.
'outskirts of' means to remove the middle letters (only letters on the outskirts).
'gijon' with its centre removed is 'gn'.
'over' going within 'gn' is 'GOVERN'.
(Other definitions for govern that I've seen before include "Exercise authority over a people" , "Control, determine" , "Rule over or regulate" , "Rule with authority, perhaps a country" , "Rule or administer" .)