First, one note about phrasing. The Civil War could not literally provide mutual aid to African Americans. This is because mutual aid is something can only be provided by people within the same group as those receiving the aid. In other words, only African Americans can provide mutual aid to other African Americans.
That said, we can say that the Civil War provided an opportunity for African Americans to give mutual aid to one another. The most important way in which this happened was through African American participation in the war. African Americans participated in the war effort in many ways. Slaves ran away from their plantations and then sometimes did things like serving as spies for the North. Others simply worked as laborers for the North. But the most conspicuous way in which blacks participated in the war effort was by serving as soldiers for the Union. They were eventually allowed to do so in late 1862. This counts as mutual aid because the war effort did a great deal to help African Americans as a group. It was this war that led to them being freed from slavery. Therefore, any actions by African Americans to promote the war effort can be seen as mutual aid being provided to others of their race.