(CNN) One week after the crushing defeat of her Brexit deal, Prime Minister Theresa May returned to Parliament on Monday to announce that she would be scrapping a controversial fee for European citizens wishing to stay in Britain, saying that it was clear that the government's approach "had to change."

May said there would be "no financial barrier for any EU nationals who wish to stay" in the UK, adding that anyone who had already paid the fee would be reimbursed. Anyone over the age of 16 seeking settled status in the UK after Brexit was due to have to pay a £65 ($84) application fee. Those under the age of 16 faced a fee of £32.50 ($42).

Guy Verhofstadt, the European Parliament's Brexit coordinator, told CNN he was "delighted the planned settled status fees have been waived."

"This is a victory for common sense. Throughout the negotiations, the European Parliament has pushed to ensure that EU citizens are not financially punished for the Brexit referendum outcome," Verhofstadt added.

May announced the decision to ditch the fee during a statement to the House of Commons about the government's next steps for Brexit, a week after lawmakers rejected her original deal by a margin of 230 votes -- the biggest defeat for any UK government in the modern parliamentary era.

May won't rule out no-deal Brexit

May said Monday that her government would continue discussions with other political parties on modifications to her deal before seeking fresh concessions from Brussels, but refused to rule out the possibility of Britain crashing out of the EU with no deal at all on March 29.

Opposition leader Jeremy Corbyn, who has refused to take part in talks until May rules out the no-deal option, said May was in "deep denial" about her "undeliverable" deal and "must change her red lines." He added that the Prime Minister's statement felt a bit like "groundhog day."

May countered that the right way to rule out a no-deal was to first approve a deal with the EU.

"When people say rule out no deal, the consequences of what they are actually saying are that if we in Parliament can't approve a deal, we should revoke article 50. Mr. Speaker I believe this would go against the referendum result, and I do not believe that that is a course of action we should take or which this house should support," May said.

And it's unclear whether it's even in the Prime Minister's remit to rule out a no-deal.

Brier added: "Saying you've got to take no-deal off the table is meaningless nonsense. The clock is ticking," Brier said. "Legislation exists which says the UK will leave on March 29, 2019. If a deal -- like May's with the EU -- is agreed, then obviously you leave with that deal. If you don't, then you leave without a deal."

May seeks backstop breakthrough

May also said that she will seek fresh concessions from Brussels on the Irish backstop, a controversial insurance policy created to prevent a hard border between Northern Ireland and the Republic of Ireland no matter the outcome of Brexit.

"All of us agree that as we leave the European Union, we must fully respect the Belfast Agreement and not allow the creation of a hard border between Northern Ireland and Ireland -- nor indeed a border down the Irish Sea," May said, rejecting media reports that she was considering amending the Good Friday peace agreement in order to pass her Brexit deal.

May began her statement to Parliament by condemning Saturday's car bomb attack by the "New IRA" in Londonderry, Northern Ireland, which politicians there have underlined as an example of just how destabilizing any changes at the border could be.

"This House stands together with the people of Northern Ireland in ensuring that we never go back to the violence and terror of the past," May said.

May's move on Monday is widely being seen as an attempt to win over Northern Ireland's Democratic Unionist party and hard-line Conservative Brexiteers by addressing their concerns over the backstop section of her plan.

But the European Union has already rejected any reopening of negotiations on the deal it took nearly two years to hammer out, never mind reassessing the details of the Irish backstop. Both the UK and the EU believe that a return to border checks could put the peace process in jeopardy, but neither have come up with a way of preventing them.

"There is no wish on the EU side to renegotiate the Withdrawal Agreement. The political declaration on the future relationship could still be enhanced in the coming weeks," Verhofstadt told CNN Monday. "On this, the door is open, but time is running out."

Without attempting to bring opponents onside, May creeps closer to a no-deal Brexit -- the UK crashing out of the EU on March 29 without an agreement having been ratified.