Claude was born in Mountain View, Kiowa County, Oklahoma. His family later lived in Texas and them migrated to Yuma, Arizona. He married twice, first to Frances/Fannie Williams about 1917, probably in Yuma. They had one child, Claude Lee Yates, born in 1918 in Yuma. Claude married again in 1919 to Carmel Shook in Imperial County, California. Together, they had at least four children, Noel, Truman, Geraldine and a stillborn daughter.
Here's a link to the family in the 1930 census.
"United States Census, 1930," database with images, FamilySearch(https://familysearch.org/ark:/61903/1:1:XH1Z-Q2P : accessed 27 April 2021), Claud W Yates, Somerton, Yuma, Arizona, United States; citing enumeration district (ED) ED 19, sheet 5B, line 65, family 90, NARA microfilm publication T626 (Washington D.C.: National Archives and Records Administration, 2002), roll 63; FHL microfilm 2,339,798.
By 1940, he had divorced Carmel and was living with his parents in Yuma.
Claude left a significant paper trail of records in Yuma and it should be easy for you to track down additional details.