To be readmitted to the Union, North Carolina had to ratify a constitutional amendment that...?

gave voting rights to all men.

Explanation:

A constitution refers to a set of written laws and principles which is typically used to determine the power and authority of the government, as well as guarantee the fundamental rights of its citizens.

A state legislature refers to the legislative body of each states of the federal government and are saddled with the responsibility of making or enacting laws. Thus, the state legislatures is one of the body of governance that has the power or authority to ratify (approve, confirm or give consent) a proposed amendment to the Constitution by getting three-quarter of the states to vote in support.

During the American Revolutionary War, North Carolina was a divided state and reluctantly joined the confederacy because the state of Virginia did so.

The ratification of the Constitution known as the fourteenth (14th) amendment which held on the 4th of July, 1868 by the North Carolina General Assembly of 1868-1869, readmitted North Carolina to the Union but required an adoption of a universal manhood suffrage for the people.

Hence, to be readmitted to the Union, North Carolina had to ratify a constitutional amendment that gave voting rights to all men without regard to their race.