Jenny Jenkins was discussed, with the version posted here, in thread 48821. Jenny Jenkins

Being a children's play version of the 1823 American (and older English-Scotch) originals, I wouldn't expect it to have any particular meaning other than the obvious about a troublesome gal.
The original (in the DT,) ends with the gal asking "So buy me a tally wally aye, sir," also meaningless, but nonsense lines used as a sort of chorus are common in 18th-19th century songs.

But read previous threads, please, and post in the thread linked above before we go around in another circle endlessly repeating what has already been discussed. Masato and Malcolm have given basic information there.