Which of these measures was specifically designed to limit the voting rights of the black freedmen?

The Voting Rights Act of 1965, signed into law by President Lyndon B. Johnson, aimed to overcome legal barriers at the state and local levels that prevented African Americans from exercising their right to vote as guaranteed under the 15th Amendment to the U.S. Constitution.

15th amendment is the answer.

the grandfather clause

its main object was the establishment of "settlement houses" in poor urban areas, in which volunteer middle-class "settlement workers" would live, hoping to share knowledge and culture with, and alleviate the poverty of, their low-income neighbors.