Ukrainian officials have said that overnight strikes by Russian missiles and drones had left at least 600,000 people without electricity.
For the second consecutive night, Ukraine said it had not received advanced warning from its allies of a wave of missile strikes coming from Russia.
Zelensky has suggested the US had threatened to restrict intelligence-sharing unless he agreed to sign up to an energy ceasefire. Trump has denied the US is holding back information.
An adviser to the president has also said Ukraine’s ability to defend itself from incoming missiles has been hindered by disruptions to its connection with the Starlink internet network, owned by Elon Musk’s SpaceX company.
Serhiy Beskrestnov said the move “cannot be a coincidence and is related to the current political situation”. The firm has not commented.
Speaking in Downing Street, Burnham said the recent days had been “dark and difficult” for Ukraine, with the country suffering “heavy bombardment”.
He told the Ukrainian leader: “You have seen the solidarity and the support you have got from everybody here in Britain.
“We are behind you and that will not change.”
Earlier in the day, Burnham had been invited to criticise the US decision to lift sanctions on Russia, when, at Prime Minister’s Questions, Lib Dem leader Sir Ed Davey asked if the PM agreed there had been a “shameful Trump betrayal”.
Burnham acknowledged that Zelensky was “carrying the heaviest of weights” but avoided attacking Trump, instead telling MPs that his role was to “repair bridges so that the support comes back”.
“Therefore I should, at this moment in time, be careful in terms of laying the ground to bring people back together.”
He added that Zelensky would not come away from his trip “empty-handed”.
In a statement following the meeting, a Downing Street spokeswoman said the leaders had discussed “Russia’s blatant indiscriminate attacks on civilians and energy infrastructure”.
She said the prime minister promised that the UK would continue to develop “innovative military solutions for use on the battlefield in Ukraine, including retrofitting British capability onto equipment gifted from other partners”.