Theresa May has scrapped the £65 fee millions of EU citizens were going to have to pay to secure the right to continue living in the UK after Brexit.
She also vowed to seek changes to the Irish backstop from the EU.
But she again rejected calls to rule out a "no-deal" Brexit - and warned another EU referendum could threaten the UK's "social cohesion".
Conservative rebels and Labour accused her of being in denial about the scale of opposition to her Brexit deal.
Mrs May had been forced to make the statement setting out her plan for how to proceed after MPs rejected her Brexit deal with the EU by 230 votes last week - her "plan B".
MPs are due to vote on a modified version of the deal next Tuesday, although she gave few details about how it would be changed.
Between now and then, MPs will table a series of amendments - proposed changes to the PM's deal - in an attempt to force through alternatives to her Brexit deal. The Commons Speaker John Bercow will choose which amendments are put to a vote.
Mrs May promised to seek the views of Northern Ireland's Democratic Unionist Party - who keep her minority government in power - and others on the proposed Irish backstop.
She said she would then "take the conclusions of these discussions back to the EU".
She said she was exploring potential "movement" on the backstop that could secure the backing of a majority of MPs, with the aim of addressing concerns that it could become permanent and threaten the integrity of the United Kingdom.
The backstop is the "insurance policy" in the withdrawal deal, intended to ensure that whatever else happens, there will be no return to a visible border between Northern Ireland and the Irish Republic after the UK leaves the EU.
Both the UK and the EU believe that bringing back border checks could put the peace process at risk.
May promises more 'flexible' approach
The prime minister attacked the Labour leader for refusing to take part in talks with her on the way forward.
She promised to take a more "flexible, open and inclusive" approach to involving MPs and the Scottish and Welsh governments, in negotiating a future relationship with the EU - once her Brexit deal has been approved.
But Labour's Yvette Cooper asked why MPs could not be given a greater say now.
"Why not put to Parliament some votes on her red lines, including a customs union, otherwise how can any of us believe a word she says?"
Conservative rebel Sarah Wollaston, who backs another referendum, tweeted that it was "like last week's vote never happened".
It’s like last week’s vote never happened. Plan B is Plan A— Sarah Wollaston MP (@sarahwollaston) January 21, 2019
What did Mrs May say about another referendum?
She told MPs: "Our duty is to implement the decision of the first one.
"I fear a second referendum would set a difficult precedent that could have significant implications for how we handle referendums in this country.
"Not least, strengthening the hand of those campaigning to break up our United Kingdom.
"It would require an extension of Article 50. We would very likely have to return a new set of MEPs to the European Parliament in May.
"And I also believe that there has not yet been enough recognition of the way that a second referendum could damage social cohesion by undermining faith in our democracy."
Earlier, MPs who back a second referendum said they had a "constructive" meeting with the prime minister's de facto second-in-command David Lidington.
What was Jeremy Corbyn's reaction?
The Labour leader accused Mrs May of being in "deep denial" about the scale of opposition to her "undeliverable" deal, which was rejected by 230 votes in a Commons vote last week.
He said Labour would back an amendment next week that would rule out the "disaster" of a no-deal Brexit - and he challenged her to confirm that she would do that if MPs voted for it.
He rejected her "phoney" offer of cross-party talks to find a way forward.
Mr Corbyn wants to force a general election and, having won it, negotiate a deal with Brussels that would see the UK in a permanent customs union with the EU, with strong ties to the single market and guaranteed protections for workers' rights and environmental standards.
If he can't secure a general election he has not ruled out getting behind calls for another referendum.
What amendments or bills have been tabled so far?
The official Labour amendment, put down on Monday evening, says MPs should be able to vote on the option of a closer relationship with Europe - with a permanent customs union - and also keep the option of a second referendum on the table.
Shadow chancellor John McDonnell said it was about "wrestling control of this process away from the prime minister into the hands of Parliament".
Labour MP Hilary Benn, who chairs the Brexit Select Committee, has tabled another amendment, calling for the Commons to hold a series of indicative votes on a way forward.
Just tabled an amendment for next week’s Brexit debate calling for the House of Commons to hold a series of indicative votes on a way forward. These should cover the four options identified in the @CommonsEUexit report published last Wednesday (attached). pic.twitter.com/i5zUFU7a8x— Hilary Benn (@hilarybennmp) January 21, 2019
Labour MP Rachel Reeves has said she has put forward an amendment that aims to stop the UK leaving the EU without a deal.
The voice of Parliament must be heard as we try to find a way forward.— Rachel Reeves (@RachelReevesMP) January 21, 2019
This evening I tabled an amendment to the Government's latest motion on Brexit to stop us leaving without a deal.
Thanks to Hilary Benn, Dominic Grieve and Norman Lamb for support. pic.twitter.com/qLxMICUS7M
And Labour MP Yvette Cooper has tabled an amendment seeking an extension to the Article 50 deadline.
She also heads a group of MPs that has, with backing from Remainer Conservative MPs, tabled a bill that would delay the UK's planned departure date from the EU if the government is not able to get its deal through by 26 February.
PM’s plan B today was just plan A again even tho that lost by 230 votes. I am really worried that PM’s drift & delay is pushing country into deeply damaging No Deal by accident. That’s why I’ve put forward this bill https://t.co/7L0oOq3fJu— Yvette Cooper (@YvetteCooperMP) January 21, 2019
Private Members' Bills - laws proposed by MPs who are not in the government - can be passed but there's normally only very limited time to debate them. The government usually controls the agenda - what gets debated in Parliament.
Some MPs now want to suspend the normal rules to allow time to debate and vote on a bill that would rule out a no-deal Brexit.
That might not be enough though. If the bill will involve spending money it also needs a "money resolution". That has to be proposed by the government.
So MPs face another obstacle if they want to take control of the Brexit process.
Why does the Irish backstop matter?
Under Mrs May's deal, if there is not a trade deal or other agreement between the UK and the EU when the transition period ends, the backstop kicks in.
It would see Northern Ireland staying aligned to some rules of the EU single market.
It would also involve a temporary single custom territory - effectively keeping the whole of the UK in the EU customs union - unless both the EU and UK agree it is no longer necessary.
But this has been a huge issue for many Conservative MPs and the DUP, who have supported Mrs May's government since the 2017 election.
Removing or amending the backstop could provide Mrs May with enough backing from Brexiteer Tory MPs and the DUP to get her deal passed by Parliament.
When will the Brexit fee be scrapped?
Mrs May's plan would mean millions of EU citizens living in the UK would have to apply for "settled status" to remain after Brexit.
Applicants must have lived in the UK for five years and had been expected to pay a fee of £65 each.
"Settled status" would give EU citizens the same access to healthcare and education after Britain leaves the EU as they have now.
Mrs May told MPs she had listened to the concerns of EU citizens - through campaign group the 3million - about the fees, which would be waived when the scheme was launched on 30 March.
The government launched a pilot scheme this week for people to apply for leave to remain through a smartphone app.
Mrs May said anyone applying during the pilot phase will have their fee reimbursed, with further details to be announced shortly.
What else did the PM promise?
In her statement, Mrs May said:
- Parliament will have "a proper say" in the next stage of negotiations on the future relationship between the UK and the EU
- the government plans to hold confidential committee sessions to ensure Parliament has the most up-to-date information
- there will be "regular updates" in the House of Commons, in particular before the six monthly review points with the EU
- she would engage "more deeply" with businesses, civil society and trade unions
- she will ensure that Parliament is provided with a guarantee that protections for workers' rights and the environment will not be "eroded"