An excellent source for this question is Ira Katznelson's "When Affirmative Action Was White." The short answer is that, no, Black Americans were not afforded the same opportunities as White Americans during the New Deal and other social programs that began after WWII (like FHA mortgage assistance). This was a function of the fact that many of the programs were administered at the state or local level, which meant that (especially in the South) Black Americans were subject to the prejudices of the local governments.

In the New Deal historiography, the primary critique of the New Deal among many Marxist and New Left historians was just as you stated: that the New Deal failed minorities. The best of these works, among the many that I'll list below, is William Leuchtenburg's 1963 Franklin D. Roosevelt and the New Deal, 1932-1940, in which he argues that the New Deal only represented a "halfway revolution" because it did not reach (or even try, in many cases) "slum dwellers, sharecroppers, and most negroes (232, 347)." Others, notably Barton Bernstein in a 1968 essay, blasted the New Deal for simply propping up or "rescuing" the capitalist structure of corporate liberalism and failing the truly Depression-stricken...especially minorities.

In a nutshell, the largest criticism from the post-1950s left regarding the New Deal is that it did little to actually serve citizens that needed it most. Even one of its flagship programs, the Tennessee Valley Authority, created to be a bureaucratic, apolitical, racially progressive beacon of the New Deal, continually ignored its own racial policies while only strengthening the wealthy farmers and business people of the Tennessee Valley via its planning/utility programs. See Nancy L. Grant's TVA and Black Americans: Planning for the Status Quo (1990) for more information about this.

This response is to serve as support for one of the previous answers that decentralized, locally-run New Deal programs reflected that area's bias toward race...not to ignore it.

See also: Ellis Hawley's The New Deal and the Problem of Monopoly: A Study in Economic Ambivanlence (1966); Bernstein's "The New Deal: The Conservative Achievements of Liberal Reform," in Towards a New Past: Dissenting Essays in American History; Michael Hiltzik's The New Deal: A Modern History (2011)