I just tried posting a fairly lengthy reply, but it didn't appear to "take" so I will summarize.

Michigan Births & Christenings 1775-1995:
Jessie Barnhart
Female
b. 05 Feb 1888
Van Buren, Wayne, Michigan
father Charles Barnhart
mother Clara B. Barnhart
Imdex # C03420-7 Michigan-ODM Film 2321097

Also a probable birth record for Del under an unprobable spelling:
FamilySearch International Genealogical Index v5.0
North America
IGI Individual Record
Adelbert Barnhord
Male
b. 07 Sep 1885 Romulus, Wayne, Michigan
Christening 14 Jun 1886 , Wayne, Michigan
father Chas Barnhord
mother Clara Barnhord
Batch # C020088 Film

Also found additional listings in 1900 census:

Romulus, Wayne, Michigan
Adelbert Barnhart, 14, b. Sep 1885 farm labor
listed as border with Ellis Bird family

Tobacco, Gladwin, Michigan
Brown, Charles, head
Clara L., wife
Dudley, son
Florence F, daughter
Barnhardt, Adelber step son
Minnie, step daughter
Margareta, step daughter

So it looks like Charles died, Clara moved back with her father & 2 youngest while Del borded (as a hired hand?), then Clara remarried & moved (reuniting Del). All of this must have happened quickly to have resulted in 3 listings!