The John Day baptised 22.9.1713 was a twin - a Mary Day was also baptised that day. Actually given as 'twins' in the register. Have you found out what happened to her?
There's also another Mary baptised 5th Dec 1714 of John and Mary.
The marriage of John Day and Mary King in Feb 1712/13 was by banns.
The marriage of John Day and Mary Haise in Aug 1713 was by licence. Have you seen this?
Findmypast have a transcription, and a link to where that came from. It's from a familysearch microfilm of these Archdeacon licences, film number 095197, though it looks as if these are only indexes again. Unfortunately the film is 'locked', which means you have to go to an LDS library to view it. I doubt there would be any more information on it. You need to see the original licence application to see if it gives any more information. it should give his occupation at least. There is no other information on the transcription on findmypast other than their names. Perhaps the originals do not survive. It might be worth asking the Oxford Record Office.
Burials for Oxford are conspicuous by their absence. OXford St Thomas registers seem to have been separate ones at this time - one for baptisms, one for marriages, and one for burials. The baptisms and marriages are on Ancestry, but not the burials. Nor on findmypast, familysearch or Freereg. The film on familysearch that has some St Thomas BMDs, says there are early burials, but I could only find some burials that started in the 1800s. You may like to have a more in depth look. Film no 808945, which is not locked, and you can look at from home when you are logged in - free, if you haven't already registered an account.
It possibly sounds like another conversation with the Oxford Record Office.
My first thought was that the John and Mary, twins of John and Mary King, baptised in Sept 1713, both died, along with their mother, and John remarried Mary Haise, albeit very soon afterwards, hence applying for a licence because he was a widower, and also because he needed to get married very quickly. They then had another daughter Mary in Dec 1714, and another son John in 1717. But looking carefully at the dates, this doesn't quite work.
The second marriage was 28th August 1713, and the twins baptised on 23nd Sept 1713. Of course this is a date of baptism and not date of birth, and there is no indication of their actual date of birth in the register. Perhaps the first wife Mary died, John remarried very quickly as he had these two babies to look after, but then they both died as well.
It's a possible scenario.
Have you found siblings for Ralph Day?
Have you looked for a will of John Day after the birth of Ralph in 1751? Or after any older siblings?
it may give some clues as to occupation, place of residence in Oxford, and other snippets of information that may be helpful in pinning down the family.