The Frank and Earnest WARE found in the 1910 census enumeration, boarding with a MURPHY family, is quite possibly your Frank with his son.
There is only one *nest WARE, born about 1905 +/- two years, in Missouri in the 1910 census.
Did Earnest take a different last name after his mother remarried? I don't find him in subsequent census records.
I'll see whether I can find birth records in that area of Missouri. OR, do you have a birth record for 'your' Ernest / Earnest? I can't see because your family tree is private.