We have no idea where Richard III wanted to be buried because he left no indication and we have no historical source that tells us. It probably wasn’t either York or Leicester.
Richard’s brother Edward IV is buried at St George’s Chapel Windsor while Richard’s wife Anne Neville is buried in Westminster Abbey. His family were originally from Fotheringhay in Northamptonshire and one of his brothers (Edmund), his mother and his father are buried in the church there.
Sometimes people died and were buried in unexpected places, and there they have stayed. A good example is the earlier Plantagenet King Edward II, who was murdered in Gloucester in 1327 and buried in Gloucester Abbey (now Gloucester Cathedral) where he still lies, since in such circumstances people were normally buried close to where they had been killed.
King Richard III made large grants to York, just as he did to many other places, and he is sometimes called ‘Richard of York’ on account of his father having been Duke of York.
However, just as in the present, in King Richard III’s time aristocratic titles did not usually reflect where their holders lived, or their loyalties. In Richard III’s case, his family actually lived in Northamptonshire, where he was born. His brother, King Edward IV made Richard Duke of Gloucester at the age of nine and, when he was in his twenties, Richard controlled many parts of eastern England north of Leicestershire and Lincolnshire for his brother. However, he still retained interests and property in Wales, East Anglia, and other parts of southern England. It would not be appropriate to claim that we know Richard III had a special loyalty or affection for any particular place.
Moreover, the Wars of the Roses were not a simple territorial war between the areas we now know as Lancashire and Yorkshire. Rather, they were about the rival claims to kingship of two aristocratic families (the houses of York and Lancaster), and their followers. Consequently, major towns and cities all over England had local wealthy families and supporters of both houses claiming the throne. This is one of the reasons why the Wars of the Roses were played out in so many different battles in different places. It also explains why some people from Yorkshire actually fought with and sided with the Lancastrians, and vice versa.