Burnham informed reporters while en route to the United Nations General Assembly in New York that he had consented to the Royal Air Force conducting air-to-air refuelling for Saudi aircraft.
The arrangement lasts for a few weeks, but Burnham stated it will be monitored continuously.
The clash between the Houthis and Yemen's Saudi-supported government has flared up again, as Houthis launch missile and drone attacks on Saudi Arabia, featuring one aimed at Riyadh on Saturday that was intercepted.
The Houthis announced a blockade on Saudi maritime traffic in the Bab el-Mandeb Strait and the Red Sea, targeting Saudi-affiliated ships and oil installations within the kingdom, the leading oil exporter globally.
The Houthis made significant progress within Yemen earlier this month by capturing the vital port city of Mokha and important Red Sea islands, which enables them to more effectively oversee maritime traffic through the Bab el-Mandeb, through which around 12 percent of global trade typically flows.
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Burnham stated he was "taking measures to protect the interests of Britain and the broader area, as Saudi Arabia has been facing assaults, anticipates possible further disturbances, and it’s essential to maintain those routes (for oil) accessible."
Britain has denounced the Houthi assaults and stated that the UK will "collaborate with allies to enhance regional stability, safeguard civilians, and facilitate humanitarian access."
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UK officials previously refrained from commenting on "operational matters" but highlighted the strong defence partnership between Britain and Saudi Arabia.
The US, which targeted the Houthis during the Biden and Trump administrations, has displayed no desire to engage militarily this time.
UK officials state that the arrangement will begin in the upcoming days, involving one RAF Voyager refueling Saudi aircraft on "defensive" operations.
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Last week, the Associated Press revealed that Saudi Arabia has requested assistance from its allies, such as France, Britain, Pakistan, and Egypt, to provide air-defence support in the region to protect against missile and drone attacks launched by Houthis and other groups backed by Iran targeting the kingdom.