Obviously, you weren't aware of the Abernathy character in the movie, because he appears in many scenes. In the scene where King appears before a city council and one of its members accuse him of being an agitator who's even despised by his own people, Abernathy stands and says to the council something to the effect of, "Dr. King has all of our confidence." And as the other poster acknowledges, Abernathy leads King and others in a prayer, amid white racists in Mississippi.
But the odd thing is that, for whatever reason, Abernathy's character is omitted from IMDb's cast list for this movie. This may prompt you to say what you said. But this omission from the movie cast list on this Web site is most peculiar, because Abernathy's character did in fact appear in a number of scenes.

I wonder why the character of Abernathy is not credited here, and why the role was so small?

Yes he is depicted.During one scene, Andrew Young says the Rev Abernathy will lead them in prayer, Abernathy then proceeds to pray.