In world mythology, are there any “bad” Santa figures other than Black Peter (Zwarte Piet) and Krampus who are meant to punish naughty children?
Kris Kringle supposedly punishes naughty children with coal. In the mythos, anyway. This is rarely, if ever, carried out by parents in reality.
I would say never but I’m sure there’s at least one dick out there who’s given their kid only coal for Christmas.
The historical Saint Nicholas is said to have punched a bishop so hard in the face he broke the bishop’s nose and laid him flat out on the ground.
We didn’t celebrate Christmas nor “St. Nick Night” growing up, but after he married my brother Kowtowed to my sister in-laws delusions and observed both with their 3 children.
When their oldest was exhibiting behavior problems they used the coal in the stocking thing on St. Nick night.
It had the effect of making the kid paranoid, suspicious, pissed off, and aware that life is about compromising ones values, pay offs and bribes. I shit you not. **I SHIT YOU NOT!!!
**
She (yes, she!) is 11 now, and I wouldn’t want to be in the business world about 15 years from now.
I AM NOT KIDDING!
Does Futurama count?
I got a lump of coal for Christmas once, as a kid. Of course, I need to mention that’s what I wanted. I collected rocks, and I already had pieces of Bituminous and Lignite coal, so I wanted a chunk of Anthracite to complete the collection. 
What about the Woodland Christmas Critters?
Are you saying there’s something wrong with my gear?
I’m surprised nobody has mentioned Nackles.
December 6th is Saint Nicholas Day, and is sort of a little pre-Christmas in the Czech Republic. Saint Nicholas (in bishop’s robes, not Santa suit) roams about with an angel and a devil talking to children. If a child has been good, saint Nicholas tells the angel to give them a little present. If a child has been bad, Saint Nicholas tells the devil to punch the child in the throat.
(Okay, not really. The devil just threatens to give them a lump of coal or to put them in his sack if they’ve been bad, but of course everyone’s been good.)
In Russia, Father Frost has a grand daughter who oftne plays some pretty mean tricks on children. I remember she freezes the hands of some children who have been bad.
Did you ever get any lignite?
If you want a rather dark take on the Santa thing, there’s “The Boy Who Laughed at Santa Claus”, by Ogden Nash.
JUST the right amount of time between the OP and this. It’s all about the timing. And the random. The random and the timing. I laughed.
/Gotta queue up Badder Santa for the holidays.