Centre's beginnning to become rowdy in Yorkshire town (13)
I believe the answer is:
middlesbrough
'yorkshire town' is the definition.
'Middlesbrough' can be an answer for 'town' (I've seen this before). I also know that 'yorkshire' relates to this answer.
'centre's beginnning to become rowdy' is the wordplay.
'centre's' becomes 'middles' ('middle' can be a synonym of 'centre').
'beginnning to' suggests taking the first letters.
'rowdy' becomes 'rough'.
The first letter of 'become' is 'b'.
'middles'+'b'+'rough'='MIDDLESBROUGH'
'in' is the link.
(Other definitions for middlesbrough that I've seen before include "Port at the mouth of the Tees" , "Teesside town" .)