Sen. John McCain, back from a Thanksgiving visit to Iraq, said Monday there are signs of progress though the country remains mired in corruption and U.S. commanders anticipate increased violence from a desperate al-Qaida.

The Republican presidential candidate said Iraqi Prime Minister Nouri al-Maliki "was more upbeat than I've seen him in the past." Still, McCain said he and other members of a congressional delegation made it clear that al-Maliki needed to show political advances.

"So we'll see what the Iraqi government does," McCain said at a lunch with reporters at his campaign headquarters in Arlington, Va.

He had been a vocal critic of former Defense Secretary Donald Rumsfeld's strategy in Iraq and had called for the United States to increase the number of troops in Iraq. Earlier this year, following Rumsfeld's resignation, the Bush administration did boost its military presence, and violence has been reported to have decreased somewhat.

McCain has used his stance on the war to distance himself from Bush while at the same time making note of the gains under the current troop surge.

He said Democratic presidential candidate John Edwards, in joining other Democrats critical of the troop increase early this year, had referred to the shift as the "McCain strategy, the McCain surge."

"He doesn't anymore, but I wish he would," he said, chuckling.

At another point, he said: "I'm the one that got criticized by Republicans because I had no confidence in Rumsfeld, that I thought the strategy was failing and that we ought to have a new strategy, None of the others who are running for the Republican nomination, much less the Democrat, pointed this out."

Despite improvements in Iraq, McCain said top U.S. commander Gen. David Petraeus was expecting a new round of violence because al-Qaida was being flushed out of its strongholds.

"He thinks they are not finished," McCain said.

McCain was on his way to South Carolina, where military troops and veterans make up a sizable proportion of Republican voters. South Carolina Sen. Lindsey Graham, an ally of McCain who traveled with him to Iraq, sent out a fundraising appeal setting a goal of $380,000 by Friday.

McCain, an Arizona senator, offered reasons for both an optimistic and a pessimistic view of conditions in Iraq. He said local city councils were beginning to govern effectively and members of the Sunni minority were being integrated into the military and were obtaining jobs. But he said corruption was still rampant in the country, noting that oil destined to Iraq's northern regions was being illegally siphoned off.

His cautious assessment contrasted with his remarks in March when he participated in a heavily guarded visit to a Baghdad market just days after saying it was safe to walk some city streets, drawing ridicule.

In his 45-minute talk with reporters, McCain tried to reproduce the tone of his campaign bus tours, where he promotes "straight talk" in conversations with journalists.

Among his points:

— "I have to do very well in New Hampshire," he said. He said independent voters, an important part of his support, would be more likely to participate in the Republican primary there on Jan. 8 if Hillary Rodham Clinton had won the Iowa caucuses five days earlier and thus made her Democratic nomination seem more inevitable.

— The subject of immigration still dogs him, he said, particularly in South Carolina. And he said Iowa, a state he ignored in his 2000 presidential bid, is a challenge for him. "Iowa we have a great deal of work to do," he said.

— He said he respected Republican rivals Mike Huckabee, Rudy Giuliani, Duncan Hunter and Fred Thompson. Asked whether he could say the same for Mitt Romney, the former governor of Massachusetts, McCain said he simply did not know him well enough to make a judgment.