I live in Beverley… It’s market day on a Sat so that is nice to have a wander round. There are a couple of pretty churches to wander round too if that floats your boat.
Cafe Velo – Superb cycling centric cafe and the food is good
Nellies – great dingy old boozer with gas lights coal fires etc. very atmospheric but not ideal if you have kids with you.
Westwood – big ole stretch of common land that is pretty and nice to have a walk on
The Westwood Restaurant – Superb place for a nice lunch. Not the cheapest but very good – I am biased though.
Softplay – if your kids are very young then there is a softplay centre. Personally I hate them but if you need to kill time.
Leisure centre – Pool, skatepark etc. Decent size and reasonably priced.
In fairness there is a fair amount just to walk around. The town is pretty and the shops are generally good. Sadly the Army Transport Museum is gone as that was a superb museum to walk around.