(Updates with White House adviser returning to help with healthcare)
WASHINGTON Dec 7 (Reuters) - President Barack Obama, battered by weeks of turmoil over the malfunctioning HealthCare.Gov website, turned to a fresh item on his agenda on Saturday as he pressured Republicans in Congress to extend benefits for jobless Americans.
It was a sign Obama may be slowly turning the corner from one of the worst crises of his five years in office, emerging bruised and weakened from the troubled rollout of his signature healthcare law, even as big challenges remain.
"For decades, Congress has voted to offer relief to job-seekers - including when the unemployment rate was lower than it is today," Obama said in his weekly address. "But now that economic lifeline is in jeopardy." The unemployment benefits expire at year's end.
Attending memorial services in South Africa next week for anti-apartheid leader Nelson Mandela and then launching into holiday season will also allow for a change of subject from the healthcare controversy.
Obama was buoyed too by news on Friday that the U.S. jobless rate hit a five-year low of 7.0 percent.
But Republicans say the glitch-prone HealthCare.gov website is only a manifestation of a deeply flawed healthcare law in which many more Americans stand to see health insurance plans canceled and to encounter sticker shock when they sign up for Obamacare.
"So by canceling your insurance, despite a promise to let you keep your plan, the Obama administration is essentially saying it knows what's best for you and your family," said Republican Representative Renee Ellmers of North Carolina in her party's weekly address.
The healthcare law, which was passed in 2010, aims to make affordable healthcare insurance available to millions of people who have no coverage.
The Obama administration's next challenge is convincing hundreds of thousands of Americans needing insurance by Jan. 1 to give the website a try before Dec. 23. Officials must make sure the site can support the traffic, and fix problems on the back end where the website transfers enrollment information to insurance companies.
"The website was always going to get fixed," said Republican strategist Scott Reed. "But the looming problem are the thousands of people who have been dropped from their healthcare because of this government-run solution. That's what's starting to build up out there as a tidal wave."
'NOT ALL FIXED YET'
While senior White House aides are cautiously optimistic that improvements to the website are removing many of the glitches, they know the problems are far from over.
"Things are better because we're making that thing work," said a senior official. "It's not all fixed yet."
The White House said on Saturday it was bringing back its former legislative director, Phil Schiliro, for a short-term appointment to "help coordinate the policy implementation" of the healthcare law.
"We are focused like a laser on reducing health care costs implementing the Affordable Care Act, and Phil will be vital to ensuring it is done right," Obama's chief of staff Denis McDonough said in a statement.
Obama's job approval rating remains low, taking a hit from the healthcare woes and the earlier budget battle with congressional Republicans that led to a government shutdown and a close brush with a debt default.
The latest Reuters/Ipsos poll put Obama's approval rating at 38 percent, against 55 percent who disapprove, among the lowest rankings of his presidency.
Ipsos pollster Julia Clark said a recent White House foreign policy achievement, an interim deal aimed at containing Iran's nuclear program, was not registering in the polls.
"It's not a great approval number and things like Iran don't resonate enough for the American people to fuel a recovery," she said.
In a sign he wants to show his agenda has not been stalled by healthcare, Obama has taken on a new tone. After frequent apologies for the website's woes, he no longer dwells on them in his public remarks, giving them only a brief mention before emphasizing the broader benefits of the law.
He cites the numbers of Americans being helped by gaining access to free mammograms or saving on prescription drugs.
"It is these numbers - not the ones in any poll - that will ultimately determine the fate of this law," he said in a speech on Wednesday that focused on income inequality and other items on his economic agenda.
Democratic lawmakers, who have often felt slighted by the Obama White House, credit his team with extensive outreach over the healthcare law, but there are lingering strains.
Many Democrats remain frustrated the administration was not more prepared when the website rolled out on Oct. 1 and they worry about the "back-end" problems with the site.
Representative Carol Shea-Porter, a New Hampshire Democrat, voiced her concerns at a Tuesday meeting on Capitol Hill where Obama administration officials updated lawmakers on the healthcare developments.
"They do understand that time is running out and people want to know exactly what it's going to look like on Jan. 1. I know that they're speaking in good faith and that they're working very hard on it."
Representative Jim McGovern, a Massachusetts Democrat, said he believed Obama's job approval rating would rise once the HealthCare.gov site improves.
Democrats need that to happen to improve their prospects in congressional elections next November, when they will be seeking to keep control of the Senate and regain a majority in the House of Representatives.
"We've had a rough couple of weeks with the bad rollout of the Affordable Care Act," said McGovern. "But I think things are changing on that." (Additional reporting by Roberta Rampton, Caren Bohan and Jeff Mason; Editing by Peter Cooney and Vicki Allen)