These are, shall I say, at the less inexpensive end of the market in York.
Roots has been mentioned – and rightly so, it’s excellent.
Arras – not that cheap but good. https://arrasrestaurant.co.uk
Buongiorno – out of the centre but easy on a bike, and extremely good Sardinian not Italian restaurant http://buongiornoyork.co.uk
The Star in the City is reasonable quality food and fantastic quality views. http://www.starinnthecity.co.uk
Le Couchon Aveugle – translate it. I liked it others might not.
https://www.lecochonaveugle.uk
Walmgate has a few interesting and probably less expensive places.
Spark is a nice area at lunchtime normally – a food court done by hipsters.
Skosh is good. Le Couchon, Roots, Arras and Buongiorno are probably better.