Mrs Duck points out that near the M62/A1 there is a playground at Pontefract Park inside the horse racing track.
If you head via Thirsk and then East to Pickering, there are options including cafes in Thirsk.
At the top of Sutton Bank there is a visitor's Centre with thigs to look at. Maybe too old for youngster, but maybe some suitable?
Helmsley is a nice market town half way across. I suspect you could find a play area there - try google street view.
Quack!
Geoff the Duck.