This may be Sarah and John P Simpson, on the census below. Their son, John P is of the correct age? If this looks like the correct family, you might try plugging in his siblings names/dates/locations to see if you can find more records using the free search tool at: https://familysearch.org/search. Or try getting the death cert from the town clerk's office to see if it contains more information. I couldn't find the birth records for the children listed. See the links at the bottom, a John P b abt 1843 MO - with son James and or wife Cornelia (? widow White) are on census data also in Harrison, MO.
U S Census, 1880 Grant, Harrison, MO
Citing this Record: "United States Census, 1880," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/M6XQ-BNJ
John P Simpson Self M 37 MO
Sarah J Simpson Wife F 34 OH
Alice Simpson Dau F 8 MO
William M Simpson Son M 7 MO
Audrainne Simpson Dau F 5 MO
John P Simpson Son M 4 MO
Emma Simpson Dau F 10 MO
Missouri, Deaths and Burials, 1867-1976
Citing this Record: "Missouri, Deaths and Burials, 1867-1976," index, FamilySearch (https://familysearch.org/pal:/MM9.1.1/FDN9-N5S
Name: Sarah Simpson
White Female Married
Death: 08 Oct 1883 Ridgeway, Harrison, Missouri
Age: 36 Birth: 1847 Birthplace: OH
Reference ID: p1
This may be John P Simpson Sr., living with son James N Simpson, b. abt 1881 MO.
John P Sr. may have remarried widow Cornelia White?
PS No need to apologize for the contents of your question, it contains so much more useful detail than most!!!