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President Obama, vacationing in Martha’s Vineyard, spoke by phone Tuesday with Prime Minister Stephen Harper about Iraq and they agreed to provide “additional, immediate humanitarian assistance, and to continue developing options to secure the safety of the civilians on Mount Sinjar,” the White House said.

Some Republicans in Congress are urging the administration to expand the new military mission against the Islamic State, saying the group’s territorial gains demonstrate that the U.S. can’t wait several months for a new Iraqi government to revamp its own security forces.

“We need much more sustained air attacks, not just in Kurdistan, but throughout the ISIS area,” said Rep. Peter King, New York Republican. “Find out where they’re deployed and go after them [and] make greater use of the Iraqi army. It could still take months before the Iraqi government is functioning.”

Mr. King, speaking on MSNBC, called Mr. Ibadi “certainly far superior to Maliki” but said the limited U.S. airstrikes will not blunt the gains by the militants.

“We’re focused on a very small part of the ISIS-controlled area,” he said. “As far as the overall area, which is larger than the country of Jordan, we’re having no impact on ISIS and they’re moving forward.”

It remains unclear how much support Mr. Maliki, who remains acting premier, has to obstruct the formation of a new administration. One senior government official said that his fears of a military standoff in the capital had eased as police and troops had reduced their presence on the streets.

“Yesterday, Baghdad was very tense,” he said. “But key military commanders have since contacted the president and said they would support him and not Maliki.”

In both Shi’ite and Sunni districts of the capital, many spoke of a sense of relief and cautious hope for change.

“I’m very happy Maliki will not be prime minister again. I hate him; he killed my sons and broke my heart,” said 68-year-old Um Aqeel as she walked in the Karrada shopping district.

Saying two of her sons had died in violence in the past year — one while serving as a soldier in the north in May — she said: “Maliki knows only the language of war and never believes in peace, just like Saddam. Yesterday when I heard he was out I felt justice has been done by God, and my two beloved sons who were killed because of him will rest in peace.”

But as Um Aqeel offered sweets to passers-by in the mainly Shi’ite area to share her satisfaction, one man, Murtadha al-Waeli, warned her angrily that she was wrong to celebrate.

“Soon you will all regret Maliki’s going,” he said. “It was he who built a strong army. Iraq will fall apart after Maliki, and we will lose the battle with the terrorists. Shi’ites will pay a high price for losing Maliki. Just wait and see.”

In the mainly Sunni district of Adhamiya, where many people have long resented what they saw as Mr. Maliki’s determination to keep Sunnis out of positions of influence, cafe owner Khalid Saad said he hoped Abadi would learn a lesson from the past by keeping his distance from Iran and leaving Sunnis in peace.

“Maliki treated us Sunni like aliens,” he said. “We hope Abadi will learn from Maliki’s fatal mistakes and pull the country back from its sea of troubles.”

This article is based in part on wire-service reports.