A series of missteps by Republican presidential nominee Mitt Romney in criticizing President Obama’s account of the Sept. 11 attack on a U.S. diplomatic post in Benghazi, Libya, might make it harder for him to continue using the incident as the heart of his wider complaint about the incumbent’s foreign policy record.

Romney has seized on the coordinated attack, which killed Ambassador J. Christopher Stevens and three other Americans, to back his contention that Obama has weakened the United States’ presence in the world and overseen an intentional diminution of American influence abroad.

In an election likely to be decided on economic issues, Romney’s move into foreign policy has stirred tensions within his campaign. But it has also put in play an issue — management of foreign policy — on which Obama once held a commanding lead.

The presidential debate Tuesday, however, again showed the perils that Romney faces in using the Libya attack to go after the president’s leadership abroad.

He mistakenly said Obama took weeks to call the Benghazi assault “an act of terror,” even though, as moderator Candy Crowley pointed out, the president used those words in a statement he made from the Rose Garden a day after the attack.

“No acts of terror will ever shake the resolve of this great nation, alter that character or eclipse the light of the values that we stand for,” Obama said at the time. At another point in the remarks, he called the attack “outrageous and shocking,” although he refrained from using the term “terrorism” to describe it directly.

The Romney campaign contended immediately after the debate that Obama continues to offer confusing accounts of what happened in Benghazi, and the candidates’ angry exchange over Libya provided no new facts to clarify how and why the attack took place.

Before the debate, Romney spoke often about Libya, arguing that it would remain important for national security reasons, not political ones, because of the gravity of what happened. Obama’s alleged delay in using the word “terror” to describe the attack became part of Romney’s stump speech.

“It’s an issue because this is the first time in 33 years that a United States ambassador has been assassinated,” Romney told an audience in North Carolina last week. “Mr. President, this is an issue because we were attacked successfully by terrorists on the anniversary of Sept. 11. President Obama, this is an issue because Americans wonder why it was it took so long for you and your administration to admit that this was a terrorist attack.”

The Obama administration has offered shifting explanations for how Stevens and the three other Americans were killed, attributing the deaths variously to an attack that emerged from demonstrations over a YouTube video disparaging the prophet Muhammad and to a well-coordinated assault carried out by the al-Qaeda affiliate in North Africa.

In arguing for his national security record, Obama has said his use of drones, intelligence and Special Operations forces against al-Qaeda’s leadership has left the group a shadow of its former self.

But a deadly al-Qaeda attack against a U.S. mission in Libya undermines that claim to a degree.

At the time of Obama’s Sept. 12 statement, neither the State Department nor intelligence agencies had publicly concluded that the Benghazi assault was an organized act of terrorism.

But intelligence briefing papers prepared for Cabinet-level officials that day said it could have been terrorism, an assessment that shifted over the following week.

The briefing based that conclusion on the violence of the attack and the heavy weapons used, which suggested an organized assault that differed from the mob breach of the U.S. Embassy grounds in Cairo that took place earlier on Sept. 11.

“There were multiple reports that came in at the same time that were quite contradictory,” said one U.S. intelligence official, who spoke on the condition of anonymity to discuss sensitive material. “It was hard to piece together the timeline at first.”

Other officials said the verification of a planned, armed assault did not mean that a protest outside the diplomatic post did not also take place.

But the State Department said last week that the streets outside the compound were quiet all day until an explosion and gunfire at the gates about 9:40 p.m. Within moments, the compound was overrun by a large crowd of armed men, a senior State Department official told reporters.

So far, though, Libya has not appeared to be a winning issue for Romney.

Romney’s problems on Libya began when he issued a statement sharply criticizing Obama’s handling of the situation hours after the assault. His rapid response — accusing Obama of sympathizing with anti-American interests over American ones in the Middle East — drew fire from Republicans and Democrats alike for injecting campaign politics into a national security crisis in real time.

The second complication came during the vice presidential debate last week, in which Vice President Biden took the Republican nominee, Rep. Paul Ryan (R-Wis.) to task over his Libya criticism by pointing out that Congress has looked to cut funding to the State Department in a way that could affect the diplomatic security agency.

But there is no evidence that those cuts — proposed for the coming fiscal year — or the State Department’s decision not to strengthen security in Libya would have prevented the attack on the U.S. mission and what emerged publicly during a recent congressional hearing as a nearby U.S. intelligence base.

On Tuesday, Romney tried to pin Obama down on his statement that he called the attack an act of terrorism in his Sept. 12 statement, saying that “I want to make sure we get that for the record, because it took the president 14 days before he called the attack in Benghazi an act of terror.”

“Get the transcript,” Obama replied, and Crowley said to Romney that Obama “did, in fact, sir” use the word “terror” in that statement.

Several Romney advisers acknowledged privately Wednesday that Obama got the better of Romney during their Libya exchange — in large part, they asserted, because Crowley took Obama’s side.

Publicly, Romney’s advisers said the Libya upheaval lingers as a growing vulnerability for Obama in the closing weeks of the campaign. They predicted that the issue would be a major focus of next Monday’s final debate, which is on foreign policy.

“I think people see it, and it’s very troubling,” Stuart Stevens, Romney’s chief strategist, said in an interview Wednesday. “It’s going to be something that people will continue to ask a lot of questions about.”

Philip Rucker, Ellen Nakashima and Greg Miller contributed to this report.