One of the examples that springs to mind is John F. Kennedy's foreign policy. During his campaign, and even as a U.S. senator, he declaimed the "missile gap" between the United States and the Soviet Union. The Gaither Report, published in the late 1950s after the shock of Sputnik, claimed that the Soviet Union would soon have enough missiles to make a safe first strike possible against the Strategic Air Command's bomber fleet. The notion that the Republican administration of Dwight Eisenhower had allowed the Soviet Union to pass the United States in missile technology helped garner Kennedy enough support to defeat Eisenhower's vice president, Richard Nixon, in the 1960 presidential election.

Eisenhower was trapped by secrecy. The success of the U-2 program and the subsequent Discoverer surveillance satellites meant he knew there was no missile gap, and in fact the United States had a towering lead in missile technology over the Soviet Union. But those programs were so tightly classified that Kennedy was ignorant of the true state of affairs until after taking office. On Feb. 6, 1961, Kennedy's Secretary of Defense, Robert McNamara, admitted to reporters that the missile gap didn't exist. The Kennedy administration backtracked from that statement, and McNamara offered to resign, but Kennedy declined. Instead, in October 1961, Kennedy announced the true state of affairs.

Track Kennedy's speeches and actions before and after, and especially before and after the Corona satellite program returned its first really good data in July 1961. That data confirmed there were no constellations of missile sites hidden in Siberia.

Lot of talk about Kennedy and the missile gap. Another similar example happened when Reagan campaigned against Carter. The Carter administration was shutting down a bomber production line (I believe it was the B-1) and Reagan campaigned hard on the issue, accusing Carter of being weak on the Soviets. It was a populist issue too because a lot of jobs were going away as a result.

What Carter couldn't tell anyone is that he had approved the F-117 Nighthawk to replace the B-1 as the next generation of bomber and it was far superior because it was the first effective use of stealth technology. The thing was extremely classified and the longer we kept it silent the bigger our advantage would be when we finally revealed it. The world didn't learn about the existence of the Nighthawk until the first gulf war, more than a decade later.

There's a semi-famous story backing this idea, about a conversation between David Ellsberg (leaker of the Pentagon Papers) and Henry Kissinger (Secretary of State under Nixon). It's not specific to the POTUS, but applies to everyone with particularly high-level security clearances and information access.

Ellsberg found himself talking to Kissinger at a point where Ellsberg had a lot of access to highly secret documents (more than top secret), and Kissinger was about to get similar access. He warned Kissinger that upon getting his clearances, he would go through three stages.

First, he would feel like an absolute fool for several weeks as he realized that all of his previous views were missing crucial information.
Second, he would begin to doubt and ignore anything told to him by people without similar clearances, thinking that their ideas couldn't be meaningful without access to this information.
Third, he would realize after several years that the information he was getting, while secret, was still unreliable and contradictory, and that it wasn't grounds to discount other people's input.

This would have been sometime around 1968, with the Vietnam War in full swing. Ellsberg says that his goal was to try to cushion for Kissinger the impact of suddenly gaining access to huge amounts of new, secret information. He also relates that this didn't work - he met Kissinger years later after the Pentagon Papers came out, and found that Kissinger was rejecting the input of government consultants to Cambodia because "they don't have clearances".

It doesn't speak specifically to the presidency, but Ellsberg makes very clear the view that access to secret info changes the views of political leaders, not necessarily for the better. Since he was deeply involved in the Vietnam-era defense industry and eventually revealed much of this information out of conscience, there's reason to give some attention to that view.

sources:
Secrets: A Memoir of Vietnam and the Pentagon Papers, page 238
NPR's 'Dangerous Man' Daniel Ellsberg Reflects

There is one instance where this is more or less true.

When JFK was campaigning for president he made a big deal out of a purported "missile gap" with the Soviets. The reality was that even though the US only had about a dozen ICBMs in 1960 the USSR had about ... 4, perhaps as few as 2. Based on research combined with statements from Soviet officials (including Khrushchev) the official estimate was that the USSR would have had around 100 ICBMs in 1960. In reality Soviet ICBM forces didn't reach those levels until around 1963 (at which point the US had built and deployed 600).

In fact, during the campaign Kennedy was secretly briefed by intelligence officials (including the CIA Director) which disproved the "missile gap" theory, though he continued to campaign on the issue regardless. Early in JFK's administration they made use of satellite reconnaissance to assess Soviet missile capabilities, in the 2nd month the DoD came to the conclusion that there was no evidence for large scale ICBM construction efforts. In 1961 a National Intelligence Estimate came to the official conclusion that there was no missile gap (at least not one for which the US was on the wrong side of).

Well one simple and straight forward example would be Jimmy Carter pledging to release any and all information on UFO sightings to the public and scientists if he were to become president, but then decided not to, citing defense implications.

Can I take this question one step further, and ask those who know historical examples of this topic to speculate on what correlations exist with today's hot political topics? In other words, what hot political issues today may be over classified, or classified for international bargaining capital similar to the "missile gap" information from Kennedy's time? And why?

Iran's true nuclear capability? North Korea's military capabilities? China's true economic or military power? ISIS' true capability to attack the U.S.?