Paying fixed penalty notices

I'm afraid you don't have a leg to stand on. The signage, as Ed posted, is very clear and there's no excuse. There's no doubt that inconsiderate cycling on that stretch would have been a local issue once upon a time, quite possibly when through motor traffic was banned there, and it was turned into a pedestrian zone. It may be that the partial ban on cycling was introduced after there was no ban, or that a blanket, 24-hour ban was partially lifted, but if you cycled there within the hours indicated, you have to pay the fine (although I don't know if the level is appropriate, that might be worth checking; £80 seems wildly excessive).

Ruserius:

You'd think you would be able to cycle on the National Cycle Network

Unfortunately, the much-overhyped 'National Cycle Network' is extremely poor in many places, and here is probably one of those places where Sustrans have previously negotiated with the local highway authority but have not yet achieved a complete lifting of the ban. I have to say that I think the ban is justified by the way the street is currently configured, with many obstacles and fairly narrow gaps, but of course that could be changed. However, it will clearly always have a very high level of pedestrian activity during daylight hours, and with important streets like this (undoubtedly dating back hundreds, if not thousands of years as a key link), you get a lot of demand on it in different ways--e.g., as a shopping destination, and, despite the ban on through motor traffic, by through cycle traffic, that are hard to reconcile unless the street layout becomes simpler and more flexible.

Where people design 'cycle routes', you often end up with barriers like this. The agency responsible, whether they are a highway authority themselves or a charity like Sustrans, will then 'park' the issue and have it earmarked for improvement, which can take years, decades, or may never happen. They may well have information about this short dismount stretch on their web-site, but obviously people are likely to miss that completely. It's one reason why designating 'cycle routes' is rarely a good idea. The best thing is just to work on improvements wherever possible, so that you don't get the kind of cognitive dissonance expressed through the signage around that short stretch of Bridge Street.

Out of interest, I take it you're not from Peterborough? I assume you would have known about it if you were?

reticent

'reluctant'