I learned it as 'Clouting out St. Peter's Shoen', which is a fairly commonplace Lanky expression meaning 'polishing'. A 'clout' is a cloth, or sometimes clothing - 'Ne'er cast a clout till the May is out'...

I, too. understood the reference to 'The man in the Moon' as being Henry VIII.

The song is 'living', though, because I know a few modern-ish verses which have been added - such as the snail taking the mail faster than British Rail...!!