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.”