William Hemmings family can be rather confusing. In fact he says that he called all three of Cocking's half breed daughters, Agatha saying "the name of Agathas being an Appelation suitable to any of the halfbreed Ladies." So who wouldn't get confused.
However, Raymond M. Beaumont did a major job of sorting some of this out in his article
Origins and Influences: The Family Ties of the Reverend Henry Bud.
You can find this as a PDF at
http://www.google.ca/url?sa=t&rct=j&q=& ... 8099,d.dmg or you can just enter the name in Google and it will pop up.
As indicated in this article Cocking's third daughter Wash-e-soo-E'Squaw Cocking married a man named Budd. When he died she looked after herself and her children by working at the Mission etc. She did not have any children with Cook. Your notes will need a slight revision.
Cook had children with a Cree Indian woman Kahnapawanakan and married the half-breed, Mith-coo-coo-man E'Squaw "Mary" Cocking, the daughter of Matthew Cocking.
He also had children with a third woman who is the mother of Jeremiah and Charles in 1802-1806. Jeremiah's baptismal record of June 8, 1823 indicates that his mother was a half-breed woman. Charles mother is identified as "Agathas." No evidence has been found that either of these men was the son of Mary Cocking. Therefore, they are presumably children of the third half-breed Agathas, Ke-che-cow-e-com-e-coot "Betsy" Cocking.
Glad that we can work together. Wayne