(CNN) When Trump administration officials arrive next week at the United Nations General Assembly -- the high-powered conclave for global leaders to grip, grin and get things done -- the world will be watching very closely.

It's not just that President Donald Trump will make his first address to a world body he's been scathingly critical about.

Observers will also parse the team he brings, as the annual meeting once again raises questions about who actually wields influence over US foreign policy at a time when Washington remains embroiled in conflicts in Afghanistan and the Middle East, faces a nuclear crisis with North Korea, and a looming stand-off with Iran.

Traditionally a stage for the top US diplomat, this year's UNGA will feature different stars: the President himself, with Trump staying an unusually long four days in New York, and UN Ambassador Nikki Haley. Vice President Mike Pence will attend, as will economic advisor Gary Cohn, Trump's son-in-law Jared Kushner and his daughter Ivanka Trump.

Secretary of State Rex Tillerson will largely follow Trump's schedule with a smaller retinue than in previous years. The arrangement, analysts say, reflects the way the secretary of state and his agency have been overshadowed by an unorthodox administration with several foreign policy power centers and a strong predilection for military leaders.

"The perennial question will have to be asked again -- where's Rex?" said Aaron David Miller, a vice president at the Wilson Center who says he's not a critic of Tillerson's, but that he's struck by "the strangeness of what's occurring."

"If this were not a galaxy far, far away, if we were back on planet Earth, we would have the secretary of state leading on Arab-Israel issues, Iran, climate change and being the repository of responsibility for foreign policy," Miller said.

Tillerson, who the State Department said will meet with Russian Foreign Minister Sergey Lavrov at the Russian diplomatic mission in New York on Sunday, will be one of three US leaders addressing both the Security Council and the General Assembly, the other two being Trump and Pence. In his remarks to the 15-member Security Council, Tillerson is expected to focus on the dangers of nuclear proliferation, an issue that encompasses the administration's two major geopolitical concerns: North Korea and Iran.

Still, Tillerson has sometimes appeared at odds with the President over recertifying the Iran nuclear deal. White House officials have leaked Trump's irritation with his top diplomat. And other officials have created their own foreign policy fiefdoms.

Kushner has commandeered Israeli-Palestinian issues and been active on other Middle East outreach, while Haley has been the administration's most high-profile public voice on Iran and North Korea.

Tillerson is "so handicapped by the strange structure the President has created on foreign policy," Miller said, "empowering any number of people in the administration, some of whom are more connected to the President than the secretary of state."

Trump "certainly hasn't empowered the secretary of state as the primary repository of responsibility for foreign policy," Miller said.

Indeed, the US agenda for this year's General Assembly was presented by Haley and national security adviser H.R. McMaster, while one mile away, Tillerson hosted a lunch for the Community of Democracies.

The US has five priorities for the international meeting, Haley said, including UN reform, counterterrorism efforts against ISIS and other groups, humanitarian assistance, the Syrian conflict, and North Korea.

"It was interesting that Secretary Tillerson wasn't there," alongside Haley, said Stewart Patrick, a senior fellow at the Council on Foreign Relations. "My sense is that he's really in the back seat on all this. It really complicates the coherence of the administration when it speaks on foreign policy."

Tillerson himself has indicated that his highest priority doesn't lie with the administration's foreign policy challenges, but with his mission to restructure and reduce the size of the State Department.

Speaking in London to staff at the US embassy this week, Tillerson said that he hoped "we get peace in North Korea; I hope we can settle the conflicts in Syria; I hope we can settle the conflict in Libya; I hope we can develop a better relationship with Russia. But those won't be the most important things that I'll do."

"The most important thing I can do," Tillerson said, "is to enable this organization to be more effective, more efficient, and for all of you to take greater satisfaction in what you do day in and day out."

Tillerson's focus on a smaller operation might further erode his impact at the annual UN meeting, said Jon Alterman, a senior vice president at the Center for Strategic and International Studies.

He notes that Tillerson has decided to establish a smaller State Department presence at the UN meeting than in years past and notes that the former ExxonMobil CEO "seems not to rely much on his own bureaucracy."

But Alterman cautions that "UNGA is an incredibly sophisticated dance that doesn't really play to Tillerson's strengths" describing it as "speed dating from hell" -- one 15-minute meeting after another, "keeping the agenda, staying focused, working through your talking points."

"This goes on for days, and it requires a whole team to make it work," Alterman said.

State Department spokesman Heather Nauert told reporters Thursday that, "we don't feel that this year we need the bodies that we have had in years past. The secretary firmly believes, coming out of the private sector, that he needs to -- and that we all need to -- be good stewards of taxpayer dollars."

Nauert also said there's little practical need to bring support staff to New York in the numbers that past administrations have.

"There's this thing called email," she said, pointing out that "people are able to provide support staffing to our colleagues who will be in New York by emailing information in."

Tillerson will also be on Haley's turf. The former South Carolina governor took the UN position on the condition that it be a Cabinet-level post and since taking it "has been behaving as if she were a de facto secretary of state," Miller said.

He noted that while Haley delivered a speech in early September that "basically laid out the President's view" on how the US should approach its departure from the Iran deal, Tillerson has pushed to recertify the pact, putting him at odds with the White House.

Alterman's colleague at CSIS, Korea expert Lisa Collins, said Haley seems to have the ear and trust of the President. "He does seem to listen to her, at least with regards to North Korea issues," Collins said in a briefing for reporters.

Alterman notes that's true with Iran as well, arguably the second most important foreign policy challenge facing the administration after North Korea.

Initially, Alterman said, senior Iranian diplomats had hoped they could work closely with Tillerson, because of his background in the oil business.

"But in this administration I've seen a lot of Nikki Haley on Iran, and she seems to, as Lisa said, have the confidence of the President and seems to be moving forward with the agenda."

Some observers speculate that Haley's tendency to eclipse Tillerson may someday become official. "Her ambitions stretch far beyond the UN. She's a potential secretary of state, she's a potential presidential candidate," said Richard Gowan, a professor at Columbia University.

In the White House briefing with McMaster, Haley acknowledged international curiosity about the US team coming to UNGA. "They are all anxious to see what the US delegation will look like next week," Haley told reporters Friday. "There's a lot of interest in how the US is going to do, and they're going to find out we're going to be solid, we're going to be strong."