Borough of Hamlyn cut in two by beck (8)
I believe the answer is:
brooklyn
'borough' is the definition.
(Brooklyn is an example)
'hamlyn cut in two by beck' is the wordplay.
'cut in two' means to take half the letters.
'by' is a charade indicator (letters next to each other).
'beck' becomes 'brook' (both can mean a stream).
'hamlyn' with half the letters taken is 'lyn'.
'lyn' after 'brook' is 'BROOKLYN'.
'of' is the link.
(Other definitions for brooklyn that I've seen before include "part of Long Island?" , "New York area" , "Big Apple borough" , "New York City borough" , "Beckham Jr; NY borough" .)