Simply put, nope (and that was the aspect of the song that I covered in a legend years ago).
Kim recalled, "It wasn't written for anything other than to satisfy a Saturday morning animated show. 'Sugar, Sugar' was one of those songs that I kept humming. I kept loving it. It would not leave me. And with the magical mind of Jeff Barry, a great record was made. There was no time to analyze and to pontificate and to see if it made any sense. The writing and the recording and the euphoria of being part of it just excited me."
Barry also noted that it was specifically written with very little kids in mind, as that was the target of the song.
What happened is that while Kirshner kept feuding with the Monkees (especially Nesmith), until they outright fired him, he had offered them a song called "Sugar Man" and that evolved over the years into him offering "Sugar, Sugar" and Nesmith freaking out on him and punching a wall. It was all part of Kirshner's retroactive history of his experiences with the Monkees where he was the smart producer and they were morons who passed on great songs like "Sugar, Sugar."
It never happened. The incident with Kirshner and Nesmith happened in 1967 and the song was written in 1969. Completely unrelated.
The song's "video" occurred in the second season of The Archie Show, which had changed its name to The Archie Comedy Hour.
The song hit #1 and was the most popular song of 1969 in the United States.
It's an amazing piece of comic book adjacent history.
If anyone else has an interesting piece of comic book history that they'd like to see featured in the future, just drop me a line at firstname.lastname@example.org!