Theresa May has used her new year message to reassure those who voted for Britain to stay in the European Union that she will fight for their interests “around the negotiating table in Europe this year”.

In a video shot from a state room in Downing Street, the prime minister acknowledges how “divisive” June’s referendum has been and calls for unity while appearing to offer succour to those who supported the losing side. Following claims that the government is on track for a so-called “hard Brexit”, in which the UK is out of both the single market and the customs union, May insists in her message that she will seek a relationship with the EU that caters to everyone’s needs, and not just those who backed leave.

“We all want to see a Britain that is stronger than it is today,” May says. “We all want a country that is fairer so that everyone has the chance to succeed. We all want a nation that is safe and secure for our children and grandchildren. These ambitions unite us, so that we are no longer the 52% who voted leave and the 48% who voted remain, but one great union of people and nations with a proud history and a bright future. So when I sit around the negotiating table in Europe this year, it will be with that in mind – the knowledge that I am there to get the right deal not just for those who voted to leave, but for every single person in this country.”

The prime minister has said she will trigger article 50 by the end of March, launching the formal exit negotiations with the other 27 members of the EU.

However, the shape of Britain’s future relationship with the EU remains a topic on which May is refusing to provide what she has described as a “running commentary”, leading to concerns across the political spectrum about how negotiations will conclude in two years’ time.

In recent days the campaign group Leave Means Leave has demanded a “clean, swift” exit amid talk that the government could seek a transitional deal that would in the short term require the UK to abide by freedom of movement obligations, remain subject to the European Court of Justice and reduce its scope to strike trade deals outside Europe.

In her message May also returns to the claim, made on the steps of No 10 on her first day in office, that her government would prioritise the interests of so-called Jams – those “just about managing”. In an apparent reference to the fact that a high proportion of Leave voters came from less affluent groups, she says: “Of course, the referendum laid bare some further divisions in our country – between those who are prospering, and those who are not … those for whom our country works well, and those for whom it does not.”

Concluding her message, May also speaks of the late Labour MP Jo Cox, murdered last year. She says: “As the fantastic MP Jo Cox, who was so tragically taken from us last year, put it: ‘We are far more united and have far more in common than that which divides us.’”

In last year’s new year message from Downing Street, David Cameron promised “to fix the things that most annoy British people about our relationship with the EU” before a referendum.