I can't make this work at work... but
High Paradise Tea Rooms, Boltby (Bank Top), North Yorkshire (accessible by bike or on foot only!) open weekends until the schools go back, then closed over the winter before reopening in the spring. Lovely homemade cakes, scones and bacon sarnies, tea and coffee, in a lovely farmhouse setting. great place. As it's only accessible without a car, the road bike access is from halfway up boltby bank (saving you from the steepest bit), MTB (or walkers) from the Paradise Trails from Sutton Bank centre.
Lordstones Cafe, Carlton Bank, North Yorkshire, nicely re-done recently, do an excellent lunch as well as HUGE slices of cake!
Roots Farm Shop and Cafe, Rounton, North Yorkshire, farm shop who also have a very very good cafe, cyclist friendly with bike racks outside, and quite used to sweaty/mucky/wet people arriving in need of sustenance!
trying to think where else we stop....
the Ship Inn at Sewerby, good local beer selection, also do cake.
The wolds galleries and tea room, awesome cream tea, but not the most bike friendly and will ask you to sit outside (fine in nice weather!)