Race impacted the American dream because by being a certain race you were removed of opportunities that the “superior” race gets; when they try to achieve the American dream, in this case being the whites(superior race). Therefore those who weren’t in that superior race did not get the big chance to get a good education, good healthcare, a good job.. ect.. and whenever that opportunity was given to any other race other than the superior race; there was a lot of hate towards that race and lots of stereotypes being placed on them( example would be how Hispanics come here and work and then are claimed are taking jobs away from the superior race)
I’m not sure if this was the answer you were looking for but this could play a part in how the American dream was impacted by race.