On reflection I think "Richard Mounds" must really be "Richard Mounsey" because Mounsey is a good old North Country name. But what a strange song it is. The narrator, I think, woke up and totally forgot he was to be the chief sportsman at the hunt, When he showed up early in the morning someone must have told him, 'Saddle your horse and get your horn, you are in charge of this hunt!" Mounsey and Wilkinson would have been the whippers-in, and a a whipper-in would be the first to spy the fox since they ride nearest the hounds.
Note that the hunt staff is exactly right in this song: Huntsman (the narrator), whippers-in left and right, and a foot-sportsman to manage the hounds on foot.