Horrid child is vain? Not half, the French admitted: that's capital (10)
I believe the answer is:
bratislava
'capital' is the definition.
(capital of Slovakia)
'horrid child is vain? not half the french admitted' is the wordplay.
'horrid child' becomes 'brat'.
'not half' indicates taking half.
'the french' becomes 'la' ('the' in French).
'admitted' means one lot of letters goes inside another.
'vain' with half the letters taken is 'va'.
'is'+'va'='isva'
'isva' placed around 'la' is 'islava'.
'brat'+'islava'='BRATISLAVA'
'that's' acts as a link.
(Other definitions for bratislava that I've seen before include "Capital of the Slovak Republic" , "City" , "Slovakian capital" , "Capital of Slovakia" .)