The definition of "large supermarket" has changed over the years, in term of floor space, but my "tup'penneth" ...
The first recognised self-service supermarket in York would have been Hillards in Hull Road (housed in a former factory unit, which I believe was used most prominently in the manufacture of bicycles). The company was taken over by Tesco in either 85 / 86 and was run as a branch of Tesco for a short time, then (in I think this order) became: Grandways, Kwik Save, Somerfield and laterly (for the last 15+ years) a large Co-oP. As Hillards, it was probably seven aisles covering food departments downstairs, with a hardware department upstairs.
However, ASDA opened a branch sometime in the 1970's, probably three times the size of Hillards out on Jockey Lane (in a building which is now occupied by "The Range"), which was York's first recognised "Superstore", also including a cafe and petrol station. ASDA later moved half a mile up the road (1997 or 98) onto the new Monks Cross retail park, and to this day trades only a stones throw from a large Sainsbury's - built and opened around the same time. The first large planned Tesco superstore in York opened up at Clifton Moor in 1988 or thereabouts (there had been a small city centre convenience branch in the early 1980's in Piccadilly - on the other side of the road to the more recent Tesco Express, as well as their short-lived presence in the vacant premises of Hillards).
Discount supermarkets are obviously hugely popular now, and the first of those in York would have been Kwik Save, on the old Hillards site in the early 90's. We then got two Netto branches - one on Layerthorpe, built on a former scrapyard (now a small ASDA), and one up at Clifton Moor. All of those are now gone, but we have a brace of both Aldi and Lidl stores which always seem to keep busy.