Maybe not 10,000, but he had a lot more than Henry Tudor and statistically should have won the Battle of Bosworth. His 'starting' position was at the top of Ambion Hill, Henry Tudor's at the bottom, in the middle of a bog. After an initial charge down the hill and early success, he was beaten part way back by the opposing side. He was well and truly scuppered when some bloke who'd stayed out of it until that point decided to attack his army's right flank at the last minute.
It goes from Edward IV (Richard III's brother) to his son Richard, who was one of the princes in the Tower, then after a long gap to the future Henry VIII. Richard III wasn't one of them, only Duke of Gloucester. I think the one in the rhyme is the 1784 one - see the link I posted above.
I stand corrected (been through my uni notes), but he was known as Richard of York and there is some credence given to the rhyme originating from this source.
There are other sources, and no-one's completely sure.