The UK has agreed to provide defensive air support to Saudi Arabia to help counter Houthi fighters and restore stability in the Middle East while driving down the cost of living at home, Andy Burnham has said, ahead of his first meeting with Donald Trump.

The prime minister said he had agreed to a request for air-to-air refuelling of Saudi jets to defend energy infrastructure against the Iran-backed rebels and help keep oil flowing from the Gulf to ease global economic turmoil.

Burnham also revealed he will use his first meeting with Trump on Tuesday to urge the US president to de-escalate the conflict in Iran, warning that "dark clouds of global instability" are directly affecting people's lives all over the world.

The turmoil in the Middle East has pushed up global energy prices, putting the UK economy under intense inflationary pressure and increasing government borrowing costs to multi-year highs just weeks before a crucial first budget for the new government.

On the flight to the United Nations in New York, Burnham told reporters that an RAF Voyager would help refuel Saudi jets for several weeks, in his first military deployment since taking power, after the Houthis advanced to areas adjacent to the crucial Bab al-Mandab strait.

The key Gulf ally has used the waterway as an alternative route for oil exports bypassing disruption in the ⁠strait of Hormuz. The RAF operational support is in addition to the UK Sky Sabre air defence system already operating.

"We have received a request from Saudi Arabia for military support and … I agreed to that request for defensive air-to-air refuelling. On cost of living pressures and acting to secure the interests of Britain, the wider region … we need to keep those pathways open," Burnham said.

The prime minister also plans to press Trump to wind down US action in Iran, which has destabilised the region, blocked key supply chains and unleashed global economic volatility, including in the UK, where the government will struggle to drive down the cost of living as a result.

"Of course the message is that we want to see an end to the conflict as quickly as possible and we stand ready to play our part," Burnham said when asked if he would raise the issue in talks with Trump.

As well as the focus on the Middle East, other major international challenges, including the war in Ukraine and artificial intelligence, are expected to dominate Burnham's meetings at the UN general assembly.

The global security backdrop to the summit is bleak: European leaders have warned that Russia could launch a drone or missile attack on Nato territory within months to test the strength of the alliance.

Burnham's speech to the UN is expected to focus on harnessing the power of AI, amid concerns at home over the safety of the rapidly advancing technology. He will be accompanied on the trip by his AI minister, Kanishka Narayan.

The prime minister's first face-to-face meeting with Trump is a significant moment, with the US president still said to be angry over the UK's refusal to support his military action in Iran, and ongoing tensions over levels of defence spending.

Burnham will attempt to highlight the importance of the security partnership after the UK and US launched a heavyweight torpedo, designed to sink ships and deep-diving submarines, in British waters last week from a Plymouth-made drone named Excalibur.

Government insiders, however, are taking some solace from Trump's low-key response to UK sanctions on trade with Israeli settlements in the West Bank, after weeks of diplomacy with the White House over the summer.

They are also hoping to avoid a clash over energy policy. The prime minister is expected to signal to Trump that he is in favour of more drilling in the North Sea, including the Jackdaw gasfield that is likely to get the green light within weeks.

Analysts believe the balance of risk for the bilateral relationship has shifted since Keir Starmer was in office, given the Republicans face a difficult set of midterm elections and the US president's personal ratings are plummeting.

Trump is also likely to be more focused on his meeting with the Chinese president, Xi Jinping, and his address to the UN, which comes shortly before the talks on Tuesday.

Government sources told the Guardian that Burnham would try to pursue a "third way" in his relationship with the US president, neither as adversarial as the Canadian prime minister, Mark Carney, nor as publicly fawning as Starmer's early approach, and relying on the strength of his personality.

Asked whether he felt he could trust Trump, Burnham said: "Yes, of course, we trust our allies. Of course, we do." Pressed whether this applied to the US president personally, he repeated: "Yes."

Downing Street said the prime minister wanted to use his UN meetings – which will also include the European Commission president, Ursula von der Leyen – to "drive opportunity and deliver hope", making progress on areas with the potential to deliver for people at home.

Burnham has focused on domestic issues since his arrival in No 10, despite Starmer warning that it would be impossible to separate them from global affairs, given how the Ukraine war is weighing heavily on UK security and the Iran conflict on the economy.

On the flight Burnham acknowledged they were inseparable. "[Starmer] was right to say it's about how we deliver back at home and the situation in the Middle East is a very clear example of that. It is a very big driver of the cost of living pressures that we're currently facing," he said.

Before the trip, Burnham's official spokesperson said: "The British people at home are at the forefront of the prime minister's mission in New York this week. Because to deliver at home, we must ensure we are leading international efforts to brighten a darkening global picture …

"To give people the breathing space they need, he is clear we must work with our international partners to face up to the biggest challenges of our generation. So this week at [the UN general assembly] he will be leading the charge on the global issues that impact people's lives at home."

Burnham will tell world leaders the UK will focus on AI and climate change when it hosts the G20 in Manchester next year, potentially causing ripples with Trump, who has previously criticised the UK's stance on global warming.

The meeting gives Burnham an opportunity, however, to host international allies on his home turf, although the location of the summit was chosen under the Starmer administration.

Burnham has been accompanied on the trip by his wife, Marie-France van Heel, who No 10 said would be meeting her international counterparts and supporting the prime minister, whose father, Roy, died last week.