Hi Heather
The best way from Cayton to Hawes would be along the A64 to York. Keep on the A64 towards Leeds until you come to the A1 roundabout. Head north to Leeming Bar then turn left at the A684 to Bedale, then on to Leyburn, Wensley, Aysgarth, Bainbridge and then Hawes.
Whatever you do, don't go through Pickering to Thirsk, as this would mean going down Sutton Bank where caravans are prohibited.
Just remember to watch your speed on the dual carriageways on the A64 as there are sometimes bottlenecks at the end of these.
I have travelled to Cayton regularly over the past 12 months and the A64 is the best route with fewest hold-ups. I also went up the A1 to Leyburn last month and it is a good route.