An anti-Brexit protestor marches near Tafalgar Square on 28.10.2018 in London | Alex McBride/Getty Images Exclusive Poll Theresa May faces battle to persuade voters to back her Brexit deal Exclusive snap poll for POLITICO by Hanbury Strategy finds the public’s impressions of the deal are negative but there’s little appetite to topple the UK prime minister.

LONDON — Theresa May faces the fight of her political career to persuade a deeply skeptical public that her proposed Brexit deal is right for the country.

With Cabinet approval for the draft Brexit withdrawal deal agreed after a marathon five-hour meeting on Wednesday and the EU preparing to trigger a special Brexit summit to sign off on the text, attention will quickly turn to the prime minister's battle to sell the deal at home.

But according to snap polling for POLITICO by Hanbury Strategy, carried out in the hours after Cabinet's decision was announced last night, the public’s first impressions of May’s deal are negative.

Forty-five percent of those polled believe, given what they have heard about the deal, that parliament should reject it. Only 28 percent want MPs to back it, and 27 percent don’t know, according to the poll of 505 people.

The snap polling took place shortly after Cabinet approved the draft deal, and the U.K. and EU published the joint text, so it illustrates where public opinion is starting from rather than a detailed response to the deal itself.

But after weeks of political debate about the likely contents of the agreement, it is clear that the public currently sees it in a negative light.

Asked to choose between backing the deal in order to get on with Brexit, another referendum, or leaving the EU without a deal at all, just 29 percent said MPs should back the deal compared to 33 percent who favor some kind of second referendum and 27 percent who backed no deal.

May for PM

With May facing the threat of a potentially imminent leadership challenge from within her party, voters are evenly split on the fate of the prime minister. Forty-four percent said someone else should takeover, with 41 percent backing her, while 45 percent said now is not the time for a Conservative leadership challenge, compared to 40 percent who said it is.

However, not all the findings make grim reading for May. The prime minister is the clear favorite when voters were presented with a list of possible Conservative rivals — 24 percent think May is the best prime minister to oversee Brexit, compared to 13 percent for Boris Johnson, 8 percent for Jacob Rees-Mogg, 5 percent for Jeremy Hunt and 3 percent for Michael Gove.

Theresa May is also favored, by 33 to 21, over Boris Johnson in a runoff between the two for prime minister (though both lagged behind "none of the above" on 46 percent.)

Three-way split

Voters are also split on what should happen next, with a strongly pro-Brexit faction pushing for a clean break from the EU, a strongly anti-Brexit faction determined to secure a second referendum on membership, and another third of people apparently caught somewhere in between.

How this latter group comes to view May’s deal could prove critical to her fate as she seeks to persuade wavering MPs on both sides of the House of Commons that her plan is good for their constituents.

When asked to pick which of four statements best reflects their view of the deal, 29 percent of respondents to the poll agree with May’s Brexiteer rivals that the deal is a “betrayal of the referendum result.”

Ranged against this view, 27 percent agree with the argument of the People’s Vote campaign that the deal leaves the U.K. worse off than membership of the EU and “we would be better off not leaving at all.”

However, 15 percent agree the agreement is “the only deal in town and the best we’re going to get” and another 15 percent think it is “in the national interest.”

When asked what should happen if parliament rejects the deal, 32 percent think the U.K. should leave the EU without a deal, 22 percent said the U.K. should stay, and 14 percent favor a second referendum on membership of the EU. Eleven percent want a "deal or no deal" referendum, and just 11 percent back the preferred option of the Labour opposition: a general election.