There are several conflicts in "The Kind of Light that Shines on Texas" - the broader racial conflict taking place on a societal scale, and the individual conflicts between Clint at the other boys. We might argue that, while the one-on-one conflict between Clint and Oakley takes precedence to the plot and progress of the story, this may be superseded by the conflict between Clint and Marvin, between Clint and black stereotypes, and between Clint and himself; that is, who Clint is, who he wants to be, and who he thinks others should be.
This is most especially revealed toward the end of the story; Clint knows and in some way accepts that he holds racial prejudices against his own race, and he holds himself above this contempt, yet this is confounded by Oakley's inability or unwillingness to distinguish Clint from other blacks. To Oakley, they are all the same. This is why Clint is outraged that Oakley hasn't targeted Marvin, a more "deserving" victim, and why Clint demands that Oakley call him a nigger - so he can justify to himself that he, too, is deserving of it.