WASHINGTON (Reuters) - To Republican U.S. presidential contenders, North Korea’s claim that it tested a hydrogen bomb may further make the 2016 race what they dearly want it to be: a referendum on President Barack Obama’s foreign policy and, by extension, Hillary Clinton’s.

For months, these Republicans have liked to say the world is “on fire,” pinning the conflicts in Iraq and Syria, the attacks in Paris and San Bernardino, California, and the recent tensions between Iran and Saudi Arabia on Obama’s administration and Clinton’s stint as his secretary of state from 2009 to 2013.

Now, they can add North Korea to the threats they say face American voters.

“When China fell to the communists (in 1949), the question that dogged the Truman administration was: ‘Who lost China?’” said John Feehery, a Republican strategist. “The question that will dog the Democrats is: Who lost North Korea?”

The criticism on foreign policy has ratcheted up the pressure on Clinton, the likely Democratic presidential nominee in November’s election, to take a harder line on national security without handing Republicans more ammunition to argue that Obama’s stewardship has been a failure.

Analysts said Republicans may have little room to maneuver since the Obama administration’s approach toward containing North Korea did not differ materially from the one used by Republican George W. Bush’s administration before it.

“They’ve been a headache for every Democrat. They’ve been a headache for every Republican,” Michael Rubin, a scholar at the conservative American Enterprise Institute, said of the North Koreans. “North Korea may be the last remaining foreign policy quagmire that hasn’t been politicized in a partisan fashion.”

That does not mean Republican candidates did not try on Wednesday after North Korea’s announcement.

They said Obama’s foreign policy let North Korea bolster its nuclear arms capabilities, and also assigned blame to Clinton.

“Three out of the four nuclear detonations that the North Koreans have done have happened on Barack Obama and Hillary Clinton’s watch,” New Jersey Governor Chris Christie told Fox News, “and they have just not acted strongly at all around the world.”

Clinton condemned North Korea’s move as “dangerous and provocative,” and said the United States should respond with more sanctions and stronger missile defenses. She also defended her performance as Obama’s top diplomat.

“As secretary, I championed the United States’ pivot to the Asia Pacific - including shifting additional military assets to the theater - in part to confront threats like North Korea and to support our allies,” Clinton said in a statement. “I worked to get not just our allies but also Russia and China on board for the strongest sanctions yet.”

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PRESSURE ON CHINA

Businessman Donald Trump, leading the race for the Republican nomination, urged China to rein in its ally North Korea or face trade repercussions.

“China should solve that problem,” Trump told Fox News.

“And if they don’t solve the problem, we should make trade very difficult for China. ... North Korea is totally under their control. Without China, they wouldn’t eat,” Trump added.

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Texas Senator Ted Cruz blamed North Korea’s test on the “folly” of failed policies by Obama and Clinton. Cruz said as president he would “rip to shreds” the international agreement on Iran’s nuclear program and predicted if Clinton is elected in November Iran would detonate a nuclear weapon, “sadly not as a test,” over a city like Tel Aviv, New York or Los Angeles.

Kentucky Senator Rand Paul advocated drawing on China’s influence with North Korea and possibly increasing sanctions on the isolated communist state.

“There are no easy solutions,” Paul told CNN. “You want me to magically wave a wand and all of a sudden their nuclear weapons are gone?”

Paul’s remarks illustrated the bind Republicans find themselves in. While North Korea’s action may buttress their argument that it is time for their party to assume control of the White House, there is a relatively small range of policy options for their candidates to advocate, analysts said, short of calling for U.S. intervention in the region.

That was ruled out by not only the Obama administration but the Bush administration before it.

Diplomacy has been tried for years. In 2005, North Korea reached an agreement with the United States, South Korea, China, Japan and Russia to suspend its nuclear program in return for diplomatic rewards and energy assistance. Negotiations collapsed after the last round of talks in 2008, with North Korea declaring the deal void after refusing inspections to verify compliance.

“I don’t think she is going to move to advocate a military option,” Scott Snyder, who heads the U.S.-Korea program at the Council on Foreign Relations think tank, said of Clinton. “She will exploit the perception that Republicans are moving too quickly into that space. This is one (issue) where she doesn’t necessarily have to move to the right. She has the advantage of having the cliff on her backside.”