I looked in the marriage records and I found Jay Barnum's wife.
Then I found the marriage info for 3 of the kids. One of the kids was born in 1906.
But you said he was a widower by 1900.
Then I looked at the census records.
Jay Barnum was not a widower in 1900. I don’t know why his wife was not living with him in the 1900 census.
If you compare the 1900 and the 1910 you can see that the wife is back in 1910.
1900 census Michigan > Gladwin > Beaverton > District 100 > 13 Jay Barnun 27 also Albert Barnun 4 Nellie Barnun 2
1910 census Michigan > Roscommon > Markey > District 183 > 6 Jay Barnum 37 Nettie Barnum 34 also Albert Barnum 13 Nellie Barnum 11 Alice Barnum 8 Millie Barnum 4 Record shows Jay and Nettie have been married for 15 years. Record shows Nettie has had 4 kids with 4 kids still living.
Collection: Michigan Marriages, 1868-1925 Marriage Date: 23 Jan 1895 Place: Chippewa Township, Isabella, Michigan Groom - Jay Barnum, 21 years, b. 1874 in Michigan Bride - Nettie May Vanz, 17 years, b. 1878 in Ohio Groom's Father: George Barnum Groom's Mother: Unknown Bride's Father: Isaac Vanz Bride's Mother: Unknown ------ Collection: Michigan Marriages, 1868-1925 Marriage Date: 18 Jan 1921 Place: Alba, Antrim, Michigan Groom - Albert Barnum, 24 years, b. 1897 in Gladwin Co., Michigan Bride - Mildred Sparks, 16 years, b. 1905 in Lake George, Michigan Groom's Father: Jay Barnum Groom's Mother: Nettie Vany Bride's Father: Albert Sparks Bride's Mother: Luella Cowler --------------- Collection: Michigan Marriages, 1868-1925 Marriage Date: 26 May 1918 Place: Markey, Roscommon, Michigan Groom - Carol Lynn Carrick, 21 years, b. 1897 inWood Co. Ohio Bride - Nellie Barnum, 19 years, b. 1899 in Beaverton, Mich. Groom's Father: Nathan A. Carrick Groom's Mother: Pearl Gardner Bride's Father: Jay Barnum Bride's Mother: Nettie Vany ----------------- Collection: Michigan Marriages, 1868-1925 Marriage Date: 26 Oct 1924 Place: Saginaw, Saginaw, Michigan Groom - Earl Griswold, 22 years, b. 1902 in Genesee Co. Bride - Mildred Barnum, 18 years, b. 1906 in Roscommon Co. Groom's Father: Judson Groom's Mother: Mable Sherwood Bride's Father: Jay Barnum Bride's Mother: Nettie Vany ---
The difference in the spelling of Vanz and Vany is because of bad handwriting in the county clerk record books.
I hope this information will be helpful to your research