I live in nearby Whitby and am a very frequent visitor to York.
The pasty shop is one of my favourites and a regular stop especially during inclement weather.
Fresh, filling and very tasty. One failure however is that whilst nice and warm to hot most of the time about 10% are barely lukewarm.
The staff will heat the food up in a microwave but I do not like what it does to the pastry.
Better to oven heat.
There is also a good variety of other than traditional Cornish with other meats and yeeuch, veggie stuff.