If you can specify a date or time you may wish to travel then advance tickets are available from Leeds - Scarborough with First Transpennine and may reduce the fare further.
Indeed - the Advances for Leeds - Scarborough seem to be priced from £7 to £12 each way. As an example this Saturday you could travel with:
£4.10 Off Peak Day Return - Shipley - Leeds
£11 Advance Leeds - Scarborough
£11 Advance Scarborough - Leeds
Total: £26.10
Potentially, with enough notice you could make this trip for £18.10.