Which was a greater force in Georgia in the 1930s and 1940s: FDR and the New Deal or the Talmadge faction?
It would probably be more nearly correct to say that Eugene Talmadge and his backers influenced the New Deal more than exercising greater influence. Eugene Talmadge was a harsh critic of Franklin Roosevelt at a time when Georgia was a one party state. That party was the Democratic party, and in the South, (Georgia as well as other states) it was also the party of white supremacy. Talmadge's criticism was not so much that he disagreed with the New Deal as that many of Roosevelt's programs provided opportunities for Blacks. After serving his term of office, Talmadge ran for the U.S. Senate to further attack Roosevelt's policies but was defeated. His frequent rants and penchant for backroom dealing quickly caused him to fall into disfavor, even in segregationist Georgia.
Because of Southern opposition to programs for Blacks, they did not receive much help from the New Deal. The CCC and other programs were racially segregated and domestic workers (who were primarily Black) were excluded from the Social Security program. The AAA which paid for farmers to leave one fourth of their fields fallow hurt blacks as many were sharecroppers and the landlord chose their land to lie fallow so that he could collect a check from the government. This type policy was completely antithetical to Roosevelt's avowed philosophy; however he needed the support of Southern segregationists and therefore made some rather unsavory compromises.