The Whitby to Scarborough Cindertrack is supposed to be decent for a sustrans old railway line thing, never ridden it though.
I have – it’s great if you want to get some miles in with some absolutely spectacular views, not much use if you’re looking for gnar to shred. The last few miles to Scarbourough are a bit shit though, badly signposted and through a not terribly picturesque estate. There’s a pub at Hayburn Wike (think that’s how it’s spelt) that marks a turnaround point when we do it (from Whitby). Best views etc. are probably between Hawsker and Ravenscar, which incorporates Robin Hoods Bay.
And, as others have said, Dalby – about 35 minutes drive from Whitby IIRC.
Last time we were there (Whitby) me and the other half had some individual playtime on the red at Dalby, and then family rides on the cinder track.
And the moors should be rideable as they ever are at this time of year, with the dry weather. Grab an OS map and take your pick.