As Mitt Romney struggles to put a cascade of missteps behind him, the Republican presidential nominee faces a twofold challenge: first, to steer the conversation back to the economy, and second, to prevent his recent difficulties from curdling into a perception that the race is becoming unwinnable.

Republicans, although anxious, point out that polls show their nominee remains within striking distance of President Obama and that seven weeks remain before Election Day.

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But Romney’s stumbles, if they continue, could jeopardize his party’s prospects farther down the ballot. Already, the Republican Party is facing a steeper climb in its efforts to retake the Senate, and the prospect of losing seats in the House.

The latest controversy — over a leaked video in which Romney disparaged nearly half the country as Obama-supporting, government-dependent slackers — is at a minimum preventing his campaign from presenting a clear set of proposals for fixing the economy that it hoped would close the deal with the electorate.

“The challenge to the Romney campaign is, how do you make the No. 1 issue the No. 1 issue,” said David Winston, a pollster who advises the congressional Republican leadership. “Any day there are other things going on that do not allow them to make the No. 1 issue the No. 1 issue is not a good day for the campaign.”

The controversy also has afforded the Obama campaign an opening to reinforce its argument that Romney’s main interest is looking out for the wealthy, not the middle class.

“My expectation is that if you want to be president, you have to work for everyone, not just for some,” Obama said during a taping Tuesday of “The Late Show With David Letterman.”

But Obama also noted that presidential candidates slip up on the campaign trail. He expressed regret over an episode in 2008, in which a recording device caught Obama telling wealthy donors in San Francisco that some small-town Americans become bitter and “cling to guns or religion or antipathy to people who aren’t like them.”

For Romney, the time and opportunities that remain to right things are short and few.

“The danger date for Romney is October 4, the day after the first and most important debate,” said one veteran Republican operative, speaking on the condition of anonymity to talk candidly about Romney’s prospects. “And he needs to keep an eye on Capitol Hill, because members of Congress are always the first rats off the ship.”

Thus far, the phalanx of Republican forces — the party machinery, outside super PACs and grass-roots interest groups — remains in tight formation behind Romney.

But if it reaches the point where his campaign appears so damaged that it threatens the prospects of other Republicans on the ballot, that unity could be shattered. In late October 1996, for instance, the party publicly abandoned its efforts on behalf of nominee Robert J. Dole and began appealing to voters to elect Republicans to Congress as a counterweight to an inevitable second term for Bill Clinton.

Republicans began this election cycle with what appeared to be ample opportunity to regain a Senate majority. That would require winning four additional seats, or three plus the presidency, which would put vice-presidential nominee Paul Ryan in the position of breaking a 50-50 tie.

But circumstances have shifted in Democrats’ favor — most notably, with the retirement of Sen. Olympia Snowe, R-Maine, earlier this year, the primary loss for Sen. Richard G. Lugar, R-Ind., and the controversial comments that the Republican Senate nominee in Missouri, Rep. Todd Akin, made about “legitimate rape.”

All three Senate seats are now considered more likely to go or remain Democratic.

In the House, few expect Democrats to pick up the 25 seats they need to regain control. But at the moment, their campaign committee estimates that at least 28 Democratic challengers are leading or tied against Republican incumbents. And they note that a total of 66 Republican incumbents are running in districts that Obama won four years ago.

“Republican candidates who reside in moderate-to-left-leaning districts who need to keep the Democratic base pacified and still have to pull a sizable portion of independents to win: That is precisely who is hurt by this,” said Republican strategist Ford O’Connell.

Exhibit A could be Romney’s home state of Massachusetts. There, Republican incumbent Scott Brown is struggling to defend his Senate seat against a challenge from Harvard professor Elizabeth Warren.

Brown, who shares some of the same political advisers as Romney, on Tuesday denounced the comments of his party’s presidential nominee, which were made at a fundraiser in May and became public Monday when videotaped excerpts of the event were posted on the Internet by Mother Jones magazine.

In the video, Romney told a group of wealthy donors that Obama’s most die-hard supporters — whom he described as 47 percent of Americans — are “people who pay no income tax,” are “dependent upon government,” and regard themselves as “victims” who are “entitled to health care, to food, to housing, to you-name-it.”

Brown said in a statement: “That’s not the way I view the world. As someone who grew up in tough circumstances, I know that being on public assistance is not a spot that anyone wants to be in. Too many people today who want to work are being forced into public assistance for lack of jobs.”

In an interview Tuesday with Neil Cavuto of Fox News, Romney insisted that his comments were meant to describe the difference between his philosophy and Obama’s.

“Frankly, we have two very different views of America. The president’s view is one of a larger government,” Romney said. “I think a society based on a government-centered nation, where government plays a larger and larger role, redistributes money, that’s the wrong course for America.”

Romney also mentioned a 14-year-old recording of Obama that surfaced Tuesday on the Internet, in which the then-Illinois state senator spoke favorably about the redistribution of wealth during a conference at Loyola University in Chicago.

The controversy comes at the end of a turbulent week for the Romney campaign. The candidate issued what was deemed, even by some in his own party, to be an overly harsh and impulsive criticism of Obama foreign policy during the flare-up of violence in the Middle East. On Sunday, Politico published an account of internal strife within the campaign, centering on the performance of chief strategist Stuart Stevens.

Some Romney supporters predicted that, if Romney frames the issue deftly, the most recent flap could work to his benefit.

Henry Barbour, a Romney fundraiser and Republican National Committee member from Mississippi, said the videotaped remarks could “tee up a conversation that is long overdue” about “the entitlement culture that is strangling the country economically.”

“I think that this does provide a genuine contrast between what Romney wants to offer the country and what Obama wants to offer the country,” said Barbour, who is a nephew and close adviser to former Mississippi governor Haley Barbour. “Romney believes in capitalism and the free market and wants to create 12 million jobs. Obviously, I think the president is perfectly comfortable promoting government dependence.”

Robert Grand, an Indianapolis lobbyist and Romney fundraiser, said he has heard no reaction to the Romney tape from his donor network.

“It’s something a lot of your folks pay attention to, but at the end of the day we have a lot bigger issues out there,” said Grand, who said he has not viewed the tape. “This is a race for leadership in this country, and we have a man who is going to provide strong leadership. The folks I talk with are the people concerned, because for 3½ years we’ve had a president who hasn’t led.”