That sounds a bit like Père Fouettard, a.k.a. the "Whipping Father".
https://en.wikipedia.org/wiki/P%C3%A8re_FouettardOr Knecht Ruprecht - Santa's man servant - who gave switches for the parents to switch the kid with.
https://en.wikipedia.org/wiki/Knecht_RuprechtOr Belsnickel - who doesn't go with Santa, making his own trips. "he carries a switch in his hand with which to beat naughty children, but also pocketsful of cakes, candies, and nuts for good children."
https://en.wikipedia.org/wiki/BelsnickelModern stories have including a Mrs. Claus who spanks, or an elf who spanks naughty bottoms to move someone to the nice list.
I can see two types of letters. One set by people writing in about their children, coworkers, niece or nephew, inlaws, etc asking the Christmas spanker to visit someone, the other set being warning letters from the spanker to targets saying "You're on the naughty list - if you don't change I will <describe upcoming spanking>".
And thank you for the idea. Now I might have Christmas story ready for this year, or maybe next year.