The government’s decision to cap the number of unaccompanied child refugees who can be brought into the UK has been upheld by the court of appeal.

But three senior judges said there had been a breach of the “duty of fairness” in the process because those refused entry were not given any reasons for being denied permission.

The limit of 480 asylum seekers under the age of 18 was set by the Home Office after the government was forced to accept a parliamentary amendment by the Labour peer Lord Dubs.

The decision by the judges, Lord Justice McCombe, Lady Justice King and Lord Justice Hickinbottom, opens the way for children refused entry to bring individual challenges about the way their applications were handled.

The judges ruled that there was nothing wrong with the consultation carried out by the government, by which it arrived at the 480 figure.

Giving reasons for the court’s ruling, Hickinbottom said: “In my judgment, the reasons given for rejecting a child … were patently inadequate.”

Children who had their applications rejected will now be entitled to receive those reasons and can appeal against those decisions.

Rosa Curling, a solicitor at the London law firm Leigh Day, which represented Help Refugees, said: “The court of appeal said the consultation was OK so the cap stays. Only 280 of the 480 places have been taken so far.

“Those refused entry have a right to know why and therefore they can appeal against the decisions. I’m sure there will be appeals by children who have been considered and rejected.”

Dubs said: “For two years, we have fought tooth and nail for the rights of these vulnerable children. Today, I’m relieved to say that it was worth it. Every unaccompanied child we have turned away deserves an explanation and a chance to appeal the decision, and we’re delighted the courts have agreed with us on that. The decision gives some hope to thousands of vulnerable child refugees in Europe.”

A Home Office spokesperson said: “This judgment confirms that the government’s consultation with local authorities, in which they said they could provide 480 places for eligible children from Europe, was lawful.

“We continue to accept further referrals and transfers are ongoing. We will continue to work closely with participating states, local authorities and other partners to relocate eligible children here quickly and safely.”