I learned this from my mother in the 1950s. She was born in 1915 and raised in Oklahoma, and likely larned the song from radio. I remember asking when she first heaard it, but she couldn't recall. It's a classic "novelty song," and the lyrics feel like "novelty songs" of the 1930s and early '40s -- songs like "Jeepers, Creepers" and "Yes, We Have No Bananas." The reference to a Gatling gun doesn't necessarily place it during the Spanish American War or earlier. The musical style is consistent with the period between the First and Second World Wars.

Here are the lyrics my mother sang:

OH Mister Moon, Moon, bright and shiny Moon,
Hidin' behind that tree --
Well, well, well-well,
My life's in danger, but I'm scared to run,
For there's a man behind me with a Gatling gun.
Oh Mister Moon, Moon, bright and shiny Moon,
Won't you please shine down on,
Please shine down on,
Please shine down on me.