A few places to try:
1. House of trembling madness. Bonus of great local beers. Burgers, meat platters, etc… without the silly price tag. Great room. Near the minster on stonegate.
2. Meltons Too. This is the fossgate one, not to be confused with meltons (which is the expensive sister rsstaurant). Great regional yorkshire food.
3. Baileys tea room opposite the library at museum gardens. Looks nothing special but food is good. Open sandwiches are great and cheese portions are generous. And next door you can get free samples of bramble whisky, sloe vodka, spiced rum.
GBK might be alright, jamies is probably okay but expensive. The top 2 give good eating and include a bit of Yorks character.