Cats & pudding bag strings

Sing sing what'll I sing
The cat's run away with the pudding bag string
Do Do what'll I do
The cat has bitten it quite in two

I'd like to know for sure whether that is all there is to it and whether it concerned a particular type of pudding. i'm trying to write up a family history and am including a section on the things the folks used to sing or say.

Thanks for any help.

Annie B.

Are you from the U.S., Britain or elsewhere?