I thought in the original text Polly sang "Pirate Jenny" at the wedding. I don't have the script handy, but I thought she wants to perform at the wedding, she recounts a story of a scullery maid she once saw, even going so far as to have the wedding guests recite lines in which they refer to her as Jenny. I thought "Jenny" was given to Jenny so that Lenya would have more to do, she being Lenya after all.
My mind could be fuzzy, though, and though I've done a lot of research on the show (for myself, not for any greater goal), I've come up with conflicting information about various facts (or non-facts) surrounding the show.
At any rate, I thought that Polly originally had the entire "Pirate Jenny," but at some point the song was switched to Jenny so Lenya could sing it (as she does in the 1931 film and the 1954 recording). The 1994 Donmar production restored the whole song to Polly, and that's how I understand it will done in the revival as well. While Jenny does seem to get the song most of the time (such as in 1990's
Mack the Knife and the 1976 Lincoln Center production), I thought that song actually belonged to Polly.
"How do you like THAT 'misanthropic panache,' Mr. Goldstone?" - PalJoey