What's a Winnie?

In the song The Virginia Company at the beggining of the film Pocahontas, in the last verse a Winnie is mentioned in the line: With a nugget for my winnie and another one for me.

What is a Winnie?

I presume it just refers to the man's wife Winifred (short form Winnie) and that name was used in the song simply because it rhymes with Virginie in the previous verse.

P.S. You incorrectly transcribed it as, "With a nugget for my winnie". In fact, by following the link you gave, I see that the original has, "With a nugget for my Winnie" and that use of a capital letter clearly indicates it is someone's name.