First, some brief background.
Article II, Section 4 of the Constitution provides:
The President, Vice President and all civil officers of the United States, shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors.
Although that clause in Article II—which addresses the presidential executive branch—is silent on the question of whether an impeached and convicted president may become president again, Article I, Section 3 answers the question squarely:
Judgment in cases of impeachment shall not extend further than to removal from office, and disqualification to hold and enjoy any office of honor, trust or profit under the United States: but the party convicted shall nevertheless be liable and subject to indictment, trial, judgment and punishment, according to law.
Article I addresses Congress—the people's branch.
And when the people's branch has spoken and determines that a president should be impeached and removed from office, that judgment may, or perhaps inherently will, include a judgment that the president is no longer eligible for federal office.
So if the House and Senate address the issue in the articles of impeachment—we will undoubtably be done with the offending president.
The Framers of the Constitution may have been unenlightened on race. But they thought of this one.