Cycling is not permitted in 'Red Square', the red-paved pedestrian area of the centre of Abergavenny.
Even so, cycle route 42 runs through it...
An interesting blog post on cycling in pedestrianised areas:
One solution is to make the entire area 'shared use'. Another would be to restrict the prohibition on cycling to the busies times of day, eg. Mon-Sat 10am-4pm, and allowing unrestricted cycling at all other times.