These fist punches are not as forceful as the fist and nose punches which are supposed to deliver enough blow to incapacitate or stun. Here, the back legs never leave the ground. Instead, these milder punches appear to be “exploratory” in nature, possibly to get a critter to scurry through the underground tunnel so it can be heard, or to even collapse underground tunnels.
If the coyote hears movement below the surface, or feels that it is onto something, digging may follow, as in the video at the bottom. However, as seen by the first two videos, sometimes no digging at all follows the punch, because nothing was heard. In all three cases here, these coyotes came up with nothing for their efforts: either the gopher or mole got away, or maybe wasn’t even there to begin with.
Punch, then looking for movement and listening for possible activity below ground
Another punch, and then listening and looking for possible signs of life below
Here the punch is followed by digging.
Coyote Yipps 
Dec 25, 2012 @ 21:43:29
Merry X-mas Janet, keep up the good work; we your readers are lucky to have you. from the Rat’s Nest
Dec 25, 2012 @ 22:51:16
Hi Charlotte — And a Merry Christmas to you, too! The peaceable kingdom in your own back yard — with coyote Ferdinand — is my favorites: Ferdinand the Coyote, by Charlotte Hildebrand. Happy Holidays to everyone! Janet