I also had a variation of this sang to me as a child.
Though the variation I remember had the song singing to "Mr. Moon Man" not mister moon.

Mr. Moon Man, Moon Man, pretty silver Moon man,
Please shine down on me.
Oh, Mr. Moon Man, Moon Man shine your light on me.

There was a bit in the middle that I can't exactly recall, but isn't one of the variations here and then ended with:

Mr Moon Man, Moon Man pretty silver Moon Man,
Please shine down on,
Won't you shine on,
Shine your silvery light on.... me....