The email from Mr. Rhodes includes goals for Ms. Rice’s appearances on the shows and advice on how to discuss the subject of the protests that were raging in Libya and at other American diplomatic posts in the Middle East.

Among the goals that Mr. Rhodes identified: “To underscore that these protests are rooted in an Internet video, and not a broader failure of policy.” In a section called “Top-lines,” Mr. Rhodes added: “Since we began to see protests in response to this Internet video, the president has directed the Administration to take a number of steps. His top priority has been the safety and security of all Americans serving abroad.”

Earlier emails had documented the concern among White House and other administration officials about talking points produced by the Central Intelligence Agency about the Benghazi incident. The email sent by Mr. Rhodes and released on Wednesday had not been part of the previous batch of documents.

Jay Carney, the White House press secretary, dismissed the new email as irrelevant, saying that the subject of the advice from Mr. Rhodes in the email was not about Benghazi, but rather about the protests that were taking place across the Middle East at the time.

“This document, as I said, was explicitly not about Benghazi but about the general dynamic in the Arab, or in the Muslim world, at the time,” Mr. Carney told reporters. “This was part of our effort to explain our views, both as a matter of policy and as a matter of what was happening on the ground with regards to the protests that were underway around the region.”