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Volodymyr Groysman, who was Ukraine's prime minister between 2016 and 2019, said visa hassles had ripple effects and were felt by fighters in the country.
French President Emmanuel Macron also condemned Britain’s refugee effort for failing to live up to “grand statements” on helping Ukrainians.
The UK has, so far, only allowed in refugees who already have relatives in the country. On Monday, a new system allowing volunteers, who will receive £350 ($456) a month, to offer homes to Ukrainians without family ties is expected to be announced.
He said the country was “very grateful for everything the United Kingdom is doing for us in these difficult times” but also pleaded for quick action on visas.
“If there is a possibility to cut this red tape for Ukrainians and to help those people of course it will be welcome, because I think you're not going to regret the fact that you care about Ukrainians, because Ukrainians are very good people, they are very bright, they need your help,” he said.
“You also need to understand that each father or each husband who is now holding arms in his hands, he is really distracted from this war with the fact if he hears that there are some kind of technical difficulties which his family is suffering.”
Mr Macron criticised Britain's visa policy but welcomed the opening of a visa site closer to Calais and the Channel Tunnel where trains can take refugees to London.
“I welcome the British shift, which shows there was a problem,” Mr Macron said.
He said: “Despite all the grand statements … the British government continued to apply current rules that meant they did not welcome Ukrainian refugees who wanted to reach British soil saying they have to travel hundreds of kilometres in order to apply for a visa.
“I would hope that the Ukrainian men and women who have lived through horror and crossed Europe to reach their families on UK territory will be better treated.”
French Home Affairs Minister Gerald Darmanin had previously written to Home Secretary Priti Patel urging her to cut back on red tape around refugees entering the country and set up UK consular services in Calais.
On Saturday, the Home Office pointed to previous comments from Ms Patel, which stated: “We are now making the process quicker and simpler by removing the need to physically visit visa application centres for many of those who are making the perilous journey across Europe.”
In the EU, Ukrainian refugees have been granted immediate residency rights, and access to medical care, schools and jobs.