Hello Alex, I've got some info for you on this!
The railway's formal name was the Gilling and Pickering (or G&P).
It started at Gilling station and turned north from the Thirsk to Malton line at Cawton. It cut through the treacherous Caulkeys Bank with a station built some way of of Nunnington. You can still see this station from a bridge if you go on the Nunnington to Oswaldkirk road.
After sidings near Harome, the next stop was Helmsley. It was a very tight curve at the south east corner. The railway turned east with a station at Nawton (passing through Ryedale School field first!), then went through Kirkdale. The viaduct is still there and you can walk over it.
The A170 bipass was crossed and then - at last, Kirkbymoorside station was reached. The 'k' was missed out in the station name.
The station is now occupied by RUssels tractor yard.