Answer:
There were more than two groups:
Women, African Americans, Mexican Americans, Native Americans, and Caucasian Americans.
Did the New Deal work?
This very much depends on which goals we assign the New Deal. On one hand, the New Deal very much met its goal if one focuses on the parts where the New Deal provisioned for an expansion and protection of industrial workers. In that context, the New Deal very much did succeed as things like the Wagner Act and the Fair Labor Standards Act of 1938 were passed. On the other hand the New Deal outright failed to address the hardships of agricultural workers and minority workers who were affected by the Depression.
Labor protections offered to industrial workers in steel, coal, and other industries were never extended to agricultural workers and domestic workers, of which a large percentage of were African American. Likewise, the New Deal despite the massive attempts to create labor projects, did not manage to lift the United States out of the Depression. On the societal side, the better society idealized by the New Deal never came into fruition.
Instead the New Deal became an extension of what Ira Katznelson would coin "Affirmative Action for Whites." The New Deal in many ways through labor legislation, administration, and even in the disbursement of aid whether through direct methods or through things like housing loans, essentially protected the socioeconomic status of the white middle class.
By the 1940s, it was clear that the New Deal though mildly helpful in temporarily employing citizens, had done much less than the massive labor requirements necessitated by WWII. For what it's worth, the New Deal wasn't an abject failure. It did attempt to at least provide protections for workers that had long been denied rights to organize and form unions. Though the unions themselves discriminated on both race and occupation, the growth of unions enabled through the Wagner Act and the FLSA in 1938 were crucial in creating an industrialized and organized middle class who would go on to represent the general American interest in the 1950s.
Unfortunately, the New Deal's successes detracted from and strengthened the institutions which would resist against the Civil Rights Movement in the 1960s. The entrenched working middle class who had benefited massively from being a quasi-exclusive organized workforce would prove difficult to persuade even long after the passing of the Civil Rights Act in 1965.
Thank you,
Eddie