Power to choose mayor, finally, over one urban area (11)
I believe the answer is:
electricity
'power' is the definition.
(electricity is a kind of power)
'choose mayor finally over one urban area' is the wordplay.
'choose' becomes 'elect' (electing is a kind of choosing).
'finally' indicates one should take the final letters.
'over' says to put letters next to each other (in a down clue, some letters go over others).
'one' becomes 'i' (Roman numeral).
'urban area' becomes 'city' (city is a type of urban area).
The last letter of 'mayor' is 'r'.
'elect'+'r'+'i'+'city'='ELECTRICITY'
'to' acts as a link.
(Other definitions for electricity that I've seen before include "Source of light and power" , "Energy from movement of charged particles" , "Current supplier" , "Domestic energy supply" , "One form of energy" .)