There are quite a few variations in the lyrics for this song, depending on where you look on the internet, as it has been performed by many artists. Many say "And away Santy", and some also use "honey" instead of "Annie".
I suspect, however, that it may be a reference to (or at least a mutation of) the shouted reprise "Away Santianna!" of the shanty "Santianna". Another clue comes in the final two lines of the song
So sailor lads, take warning, when you land on that New York shore,/
You'll have to get up early to be smarter than a whore
So, it seems evident that the song is written from the viewpoint of a sailor, who would be familiar with the Santianna shanty.
Folk songs, shanties and the like are mostly passed on orally, rather than in writing, so mutations can easily occur, and I think that is what has happened in this case.