Almost two thirds of voters back putting an extra 1p on income tax to solve the funding crisis engulfing the NHS and social care, a new poll for the Observer has found.

The Opinium survey found that 65% would be happy to pay an extra penny in the pound ring-fenced for health and social care, even after being shown what the increase would actually mean for their personal tax bill.

It found that the NHS and health was the priority issue for voters, with 68% of voters identifying it as important. Brexit was in second place on 42%, with immigration a concern for a third (33%) of voters.

By an overwhelming margin, voters think the NHS is underfunded (77%), while most think the police (66%) and schools (59%) also require more money. When asked which party they most trust to manage the NHS, Labour has a 14 point lead (39% to 25%).

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It comes with a debate raging about how to plug the extra funding needed for the NHS. Boris Johnson has been among those pushing for a cash injection to coincide with the UK’s exit from the European Union.

Jeremy Hunt, the health secretary, has said that “significantly more” money would need to be spent on the NHS over the next decade. Meanwhile, MPs are calling for a cross-party commission to examine how to solve the long-term issues facing health and social care.

The idea of placing an extra 1p income tax to pay for health services was backed by the Lib Dems at the last election and supported by the likes of David Nicholson, the former head of NHS England.

The Opinium poll also shows that the Conservatives have pulled ahead in terms of voting intention, with a 3-point lead over Labour. The parties were neck and neck a month ago, but the Tories are now up two points on 42%, with Labour down one on 39%.

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Theresa May’s approval rating is also up slightly. The proportion of people who approve of the job she is doing has risen from 29% to 34%. When combined with the proportion who disapprove, she has a net approval rating of -11%. Jeremy Corbyn’s net approval rating also improved slightly, to -7%. Vince Cable’s net rating is currently -18%.

May has maintained her lead over Corbyn on who would make the best prime minister, with 34% choosing her and 28% choosing the Labour leader. More than a quarter (28%) said they did not know.

The poll was conducted online between 6-8 February, among 2,002 people.