You have William and Lula Pugh attached as parents of your grandfather, with a note that those names are on his S.S. application. There is a Jack Pugh, son of William and Lula Pugh, on the 1900 census in De Soto, Louisiana. But his birth info is shown as May 1893 in Louisiana. He's still in his parents' home in 1910. And I locate a Jack Pugh on the WWI draft registration cards with a DOB of 9 May 1893, and the same Jack on Find-a-Grave.
You have Lula Nobles as the mother, yet Lula appears on the 1900 census living with her parents. If Jack was born in 1898, he should be in this household. The marriage record you have attached to Lula Nobles is for a marriage to William King, not William Pugh.
You mention that the only record of Jack's parents is on a S.S. record. Can you provide a link to that? You need to be sure that it's for 'your' Jack and not the above-mentioned one that lived and died in Louisiana.
You don't go into detail about your DNA and matches. You mention you match many Pughs but you're the only DNA for your immediate family. What do you mean by that?