Iraqi Prime Minister Haider al-Abadi called on the United States and other allies to “stay focused” in the fight against Islamic State Monday, saying he sees a new willingness in the Trump administration to take a more direct role in the fight not just to contain but to defeat the terror group. “We can eliminate this terrorist organization,” Mr. al-Abadi told a Washington audience just hours after meeting with President Trump at the White House ahead of a major summit this week of officials from the 68-nation coalition battling the group also known as ISIS. “I think this administration wants to be more engaged in fighting terrorism,” he said. “I sense a difference in terms of being head-to-head with terrorism.” The Iraqi leader said recent battlefield successes in the fight to retake Mosul, the country’s second-largest city, are “proving that [Islamic State] can be killed.” “We can do it, not only in Iraq, but in the region,” he said. “I’m encouraging our allies and our friends [to] stay focused.” His comments came as Iraqi forces backed by Washington continued their slow but steady advance through the streets of Mosul on Monday — a campaign that began almost three months ago and has become a brutal street-by-street combat in recent weeks with Islamic State fighters cut off from escape. But the dangers Islamic State and other jihadi groups pose to Baghdad were also on vivid display during Mr. al-Abadi’s first Washington visit since Mr. Trump’s election: A suicide car bombing targeting a commercial area in Baghdad killed at least 23 people on Monday and wounded 45 others, according to Iraqi officials. The Trump administration has said the goal of this week’s summit in Washington will be to “increase the momentum” of the wider international campaign that the Obama administration set in motion against Islamic State nearly three years ago. While Mosul may fall during the coming weeks, there is still a larger and more complicated battle looming for Islamic State’s de facto capital in the Syrian city of Raqqa. It’s unclear how the Trump administration’s approach to the wider campaign will differ from that put in place by the Obama administration — an effort that Mr. Trump personally lambasted as a failure during last year’s presidential campaign. Mr. Trump, in his meeting with Mr. al-Abadi, hinted that a more aggressive U.S. military posture in the conflict was already underway. “Our main thrust is we have to get rid of ISIS. We’re going to get rid of ISIS. It will happen. It’s happening right now,” Mr. Trump said. Uncertainty has swirled around Mr. Trump’s overall view of the U.S.-Iraq alliance, which took a hit when Iraq was included in Mr. Trump’s original travel ban targeting seven Muslim-majority nations. After sharp criticism from Baghdad and from lawmakers on Capitol Hill, Iraq was the one country dropped from Mr. Trump’s revised executive order earlier this month. While the new order, along with the January version, remains blocked by U.S. courts, Iraqi political insiders say nerves are still raw over the matter in Baghdad. Mr. Trump also angered many in the Iraqi capital during his first days in office, when he made headlines by repeating his campaign pledges that the U.S. military may seize Iraq’s oil as compensation for having engaged in the nation for so many years. ‘Great indicators’ Lukman Faily, the former Iraqi ambassador to Washington, said he was optimistic about the Iraqi prime minister’s meeting with Mr. Trump Monday, despite uncertainty still hanging over U.S.-Iraq relations. “We have not had a lot of great indicators so far from the new administration,” Mr. Faily told an audience at the American Enterprise Institute, adding that the Trump administration’s overall posture toward Iraq has so far been “more ambiguous than we would like it to be.” “We are getting different signals and, to me, the travel ban was a clear example of how these mixed signals are being [sent],” Mr. Faily said, although he added that the relationship between Mr. al-Abadi and Mr. Trump and other top administration officials, including Defense Secretary James N. Mattis, appears “very positive.” Mr. Mattis was among a core of officials who participated in Monday’s meeting at the White House. Others on hand included senior Trump adviser Jared Kushner, National Security Adviser H.R. McMaster, Commerce Secretary Wilbur L. Ross Jr. and White House Chief of Staff Reince Priebus. President Trump seized the moment to heap praise on Mr. al-Abadi, saying the Iraqi prime minister is doing a “good job” and that Iraqi “soldiers are fighting hard” against Islamic State, particularly in the ongoing battle to retake Mosul. “Mosul is moving along,” Mr. Trump said during a few moments when reporters were allowed into Monday’s meeting. The president also reiterated his campaign trail lament that the Obama administration prematurely withdrew U.S. forces from Iraq in 2011. “Certainly we shouldn’t have left. We should never ever have left,” he said. “A vacuum was created.” But even as Iraqi forces come closer to recapturing Mosul — the main Islamic State stronghold in Iraq — there are still big questions about how willing the Trump administration will be to commit to future efforts to rebuild the country once Islamic State is defeated. President Trump’s 2018 budget proposal calls for a roughly 30 percent cut in funding for the State Department and the U.S. Agency for International Development, both of which contribute significantly to peacekeeping, development and reconstruction missions. It’s not yet clear how the cut — if passed by Congress — will affect reconstruction efforts. Some analysts have argued that Mr. Trump’s separate call for an increase in U.S. defense spending could cover any loss of Iraqi funding from the State and USAID cuts. Mr. Faily said Monday that one of the biggest difficulties for the Iraqi central government centers on bringing all of the various counter-Islamic State fighting factions in the nation — from the so-called Popular Mobilization Unit (PMU) Shiite militias to the various Sunni tribal factions — together under a single roof that answers to the prime minister’s office. “We still have the federal police, we still have the counterterrorism forces, we still have the PMUs, we still have the Army — four separate silos,” he said. “We need to align them and merge them.” Mr. al-Abadi said Monday that the Iraqi people are “proud of our security forces” and see them as “a national hero” because they “represent all Iraqis.” He also downplayed reports of human rights abuses by some of the forces in the counter-Islamic State campaign. “Our direction to them is to respect people, to respect human rights,” he said. “For you to fight for the country, you have to win over the people.” During a visit to Washington last year, Mr. al-Abadi worked to drum up greater financial and military support from the Obama administration. He also sought greater assistance to help Iraq confront a humanitarian crisis, with more than 4 million people displaced in the fighting. On Monday the Iraqi prime minister said it was a “huge accomplishment” that some 1.6 million of the displaced have returned to their homes.