This is a rather difficult question, since it is believed that there are three potential Dukes it could be referring to.
The first candidate is Richard of York, who gave battle in vain, as known from the mnemonic regarding the colours of the rainbow. In this case it would be a reference the Battle of Wakefield Green during the War of the Roses in 1460. ^[1]
The second candidate is James II, who marched his troops to Salisbury Plain in 1688 to repel an invasion from William of Orange, only to retreat them when he lost his support. ^[2]
And the third, most often cited candidate is Prince Frederick, Duke of York and Albany, second son of King George III, Commander in Chief of the British Army during the Napoleonic Wars. In this case, it would be a reference to the Flanders campaign of 1793-94, likely in or near the town of Cassel. ^[3]
SOURCES:
J. Swinnerton, The History of Britain Companion (Robson, 2005), p. 149.
C. Roberts, Heavy words lightly thrown: the reason behind the rhyme (Granta, 2004), p. 44.
I. Opie and P. Opie, The Oxford Dictionary of Nursery Rhymes (Oxford University Press, 1951, 2nd edn., 1997), pp. 442–443.