The US president earlier posted on social media that the two countries had agreed to immediately halt attacks on each other’s energy infrastructure.
Ukrainian President Volodymyr Zelensky said he was unaware of the development but that it was “good news”, while Moscow has not yet commented.
Mr Burnham said: “Russia has intensified its strikes on Ukrainian energy infrastructure over the weekend.
“This is an attack on Ukrainian civilians, plunging them into cold and darkness.
“The UK strongly supports President Trump’s call today for a sustained energy ceasefire to start now.
“I discussed this with Volodymyr Zelensky yesterday: Ukraine is ready if Russia stops its attacks. This is the best route to get energy and fuel back into world markets.”
The Prime Minister had reiterated his “complete solidarity” with the Ukrainian leader in a call on Saturday and said he plans to meet him in person in the coming days.
They spoke after what Mr Burnham called a “difficult” 24 hours for Ukraine in the wake of Mr Trump announcing a deal to obtain diesel from Russia.
Andy Burnham said he discussed an energy ceasefire with Volodymyr Zelensky (Henry Nicholls/PA)
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After Mr Zelensky objected, Mr Trump said it was “time for Ukraine to get a new president”.
Then on Sunday, the US president made another shock announcement.
He posted on social media: “Effective immediately, there is an ENERGY CEASEFIRE in the War between Russia and Ukraine. Both parties have agreed. Do not break it!”
He did not give any further details.
The Ukrainian president said Mr Trump’s announcement of a ceasefire was “good news for everyone — and it is important that it become reality”.
But he said Russia had struck a power plant in Kyiv and energy facilities in Kharkiv after the US president’s post about the ceasefire.
He reiterated his willingness to organise elections in Ukraine if Russia stuck to a 60-day ceasefire, in an interview with US broadcaster ABC.
Mr Trump, who is under pressure to bring down fuel prices ahead of November’s midterm elections that will decide control of Congress, said on Friday he had agreed to obtain millions of tonnes of diesel from Russia.
Diesel prices have rocketed in recent months because of the disruption to global supplies caused by the US-Israel war against Iran and the ongoing conflict between Russia and Ukraine.
Defence Secretary Wes Streeting sought to play down the effect the move has on US support for Ukraine as he spoke to broadcasters on Sunday morning.
He told the BBC: “We actually do a lot of procurement with and through the United States to support Ukraine with things like missile defences.
“There is clearly a big effort being led by the United States on the diplomatic track.
“So, let’s not overstate the impact of President Trump’s decision on purchasing oil from Russia, and let’s not also understate US support for Ukraine.”
The UK has pledged to maintain pressure on Russia through tough sanctions after the move from Mr Trump, which comes after the US leader signed a sweeping Russia sanctions law just last month aimed at clamping down on the oil and gas revenues that fuel Moscow’s war machine.
Donald Trump has said that Ukraine needs to get a new president (Liam McBurney/PA)
PA Wire
Mr Zelensky had criticised Mr Trump’s deal, saying it “plays into Russia’s hands” and would prolong the war.
The US president, meanwhile, lashed out at the Ukrainian leader on Saturday, suggesting Mr Zelensky had not done enough to end the war with Russia.
Speaking at the White House, he said: “I suggest they get a new leader who can make a deal″, adding that “for some reason (Zelensky) never does”.
He added: “I think it’s time for Ukraine to get a new president.”
Elections have been suspended in Ukraine since the war broke out in 2022 and the country was put under martial law.
Conservative Party leader Kemi Badenoch said Mr Trump was “wrong to ease sanctions on Russian diesel, wrong to blame President Zelensky for prolonging Putin’s war and wrong to undermine a nation fighting for its right to exist”.