From the flap:
"Does anyone but Chuck Masterson and David Topman and Mr. Bass know about the Mushroom Planet? Well, there's Mr. Tyco Bass's cousin, Mr. Theo. He is a Mushroom Person, like Mr. Tyco, so he knows. And of course David and Chuck told their families about The Wonderful Flight to the Mushroom Planet (along with a good many thousands of readers).
But what if just an ordinary human being should happen to find out about it? Would it ruin everything? The answer is in this second story about Basidium, the small planet whhich can only be seen when Tyco Bass's special filter is affixed to the telescope.
David and Chuck returning to Basidium in their new space ship, have considerable difficulty carrying out Mr. Bass's wish that the planet be kept a dead secret. One Horatio Q. Peabody makes this trip even more of an adventure than the first one."