Half of people near British cities cast vote (10)
I believe the answer is:
plebiscite
'vote' is the definition.
(plebiscite is a kind of vote)
'half of people near british cities cast' is the wordplay.
'half of' indicates taking half.
'near' says to put letters next to each other.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'cast' indicates an anagram.
'people' halved is 'ple'.
'cities' is an anagram of 'iscite'.
'ple'+'b'+'iscite'='PLEBISCITE'
(Other definitions for plebiscite that I've seen before include "Public vote" , "General vote" , "Popular vote" , "Referendum" .)