I was delighted to see that so many had grandparents who sang this song to them. I did a search and wanted to find out more about the history. Seems everyone has their own words. Mine is almost the same I'll give you what I learned:

Mr. Moon, Moon, pretty silver moon;
won't you please shine down on me?
Oh! Mr. Moon, Moon, pretty silver moon;
come out from behind that tree.
You're in danger you'd better run;
Here comes Johnny with his big-bad gun.
Oh! Mr. Moon, Moon, pretty silver moon;
won't you please shine down on me?