Jesse Jimmy is one of the ugliest "cute kid" characters I've seen. Was he based on a puppet or something?
Jesse Jimmy certainly looks like a puppet. I've never seen a TV or radio and film puppet that looked like him. But Puppet TV shows weren't generally national (other than "Howdy Doody", and Rootie Kazootie, nor international. I was in Canada all the years of Magazine Enterprises' "Cowboys 'N' Injuns" (1946-1952). So, I wouldn't have had a chance to see a local one-city TV station's puppet show, to see the puppet model for Jimmy Jesse. But, I'm guessing that Jimmy Jesse WASN'T based on a specific TV show puppet, but was a character made specifically for The "Cowboys 'N' Injuns" comic book series, and simply patterned after the looks of a generic puppet, to interest little kids.
Avon's "Sparky Smith" (from "Television Puppet Show"), must have been a local one-city puppet, like Chicago's "Garfield Goose", as there are references to the TV show in the books' intro page. But, I never heard of that show from any other source. So, I doubt that that TV program was syndicated nationally.