In Help! the Beatles are introduced via a film in which they perform the title song. Clang, upset that Ringo is in possession of the ring that's integral to his cult's sacrifices, throws darts at the Beatles' images on the screen:
In "Here Come the Monkees," Mike also throws darts at a picture of the Beatles. Ringo seems to be his target too:
Taken by itself and within the fictional context of the show, Mike's throwing darts at the Beatles illustrates the Monkees' frustration that the Beatles are so much more popular than they are. If the reference is recognized, however, this seems to illustrate how significant of an influence the Beatles were to the creators of the show, who copy even this self-deprecating element of Help! Or perhaps this similarity is just coincidental.