I live in the middle of a deep woods and have killed 5 in the yard within the last 6-7 years. Plus many affield.
Have surely seen their destruction first hand in both wildlife...and particularly with livestock.
While I have always tried the best I can to kill any animal as humanely as possible, if I see a yote wherever, don't care if I have nothing but a slingshot on hand, you can bet that just as sure as God made little green apples, there is incoming in that yotes near future.
In response to a previous post referring to 'Gods creations'...here's a couple other thoughts along those lines.
First, he also created man, putting man as the rulers and 'steward' of all other beasts. 'Steward' means man keeping the population of all other predators (including coyotes) in check so the existence on His number one creation, 'man' can survive. To those familiar with the yote explosion over the years due to less trapping and the increasing destruction of the yote, it doesn't take a rocket scientist to figure out that 'man' has failed miserably in his responsibilities to keep the population of the yote down.
Second, when all this was created, obviously man was not equipped with weaponry of today and killing these predators with spears, crude traps, pits with the bottoms laced with sharpened sticks...or whatever means they had, was the way it was done. And we can be assured that the death of the animals was most likely not near as humanely as it is today.
Like most rural areas, especially areas having livestock, around these parts yotes are a real problem that is not slowing down in the least.
And I for one would challenge anyone with a different mindset.
Also, yotes are getting to be an issue in many suburban areas as well. There have been many recent reports of incidents between people walking their dogs and yotes in neighborhoods.
Lastly, it's common knowledge that purely hunting yotes will not keep their population down. And with very few trapping, the problem is not only going to continue, but at yotes yearly multiplying rate, will do nothing but increase.
Given those facts...again, I will try as best I can to kill a yote humanely. But whatever means I have at the time to kill one is what I'll use.
As far as what went wrong with the OP's shot?
I think JohnKSa has the right answer.
You need to pattern your shotgun knowing the distant your shot pattern starts breaking up badly causing a less than effective kill.