Exclusive: Large majority of voters say hard line on Europe and immigration would make Conservative party more appealing

Conservative demands that David Cameron tack right and return to traditional Tory values will be emboldened by a Guardian/ICM poll which finds the public overwhelmingly believes a hard line on Europe, immigration and traditional families would make the party more appealing.

After a month in which the Eastleigh byelection defeat was followed by speculation about challenges to the prime minister's leadership, Cameron will be relieved his party has clawed back some of Labour's 12-point lead last month. Ed Miliband's party has slipped two points since February, to stand at 39%, while the Tories have climbed two to 31%.

Buoyed by victory in Eastleigh, the Lib Dems are up two, to 15%, Ukip drops from 9% to 7%, and minor parties remain on 8%.

But the Tory right's advocacy of the sort of policy positions associated with Ukip will be encouraged by the poll. By a majority of 67%-25%, voters across the spectrum say "the Conservatives would be more appealing if they took a tougher line with Europe".

An even larger majority say the same about "a tougher line on immigration": 75%-21%. In what the right may seize on as a rejection of gay marriage, voters judge by 69%-24% that the Tories' appeal could be boosted by keeping "themselves on the side of traditional families".

There is evidence people may be ready for a right-leaning approach to social affairs. By 65%-30%, for example, voters reject the suggestion that it is wrong for the rich to opt out of the NHS and buy their own medical care, and by 87%-10% they report being proud to be British.

On social security, following George Osborne's legislation to hold down benefits below the rate of inflation, voters are slightly more forgiving: by 50%-35% they believe workless claimants are mostly "unlucky rather than lazy". But amongst the very youngest voters, there is a 48%-47% tendency to regard claimants as lazy rather than unlucky.

This "back to basics" agenda finds apparent support across the political spectrum: on Europe, for example, the right is solid in believing a tougher line would boost the Tories (83% of Tories, and 100% of Ukip supporters), but even among Labour and the Lib Dems majorities of 56% and 64% respectively take the same view.

On immigration, the 88% of Tories and 98% of Ukip voters who believe a harder line would help the Conservatives are matched by 60% of Labour supporters and 68% of Lib Dems. And on siding with "traditional families", the 84% of Tories who believe this would help their party are matched by 62% of Labour and Lib Dem voters.

Miliband has recently dedicated two speeches and a party political broadcast to immigration – but Tory modernisers warn that, while the public inclines to a hard line, it will recoil from any party that appears obsessive.

A week before the budget, the poll finds less support for the right's chief economic demand, aired by former defence secretary Liam Fox on Monday, of making a priority of lower taxes.

After rival interventions last week by the Lib Dem business secretary, Vince Cable, who floated the possibility of borrowing in order to invest, and Cameron – who resolved instead to stay the course with austerity – ICM finds voters split down the middle between the two positions.

"Investing to kickstart the economy" is supported by 36%, against 34% who prefer doing "whatever it takes to reduce the deficit" – the coalition's "plan A". That contrasts with 21% who hope for "lower taxes" above all else.

On the question of austerity, Tories prefer deficit reduction (52%) over debt-financed investment (27%), whereas the latter option is more appealing to Labour supporters (47%) and to a lesser extent Lib Dems.

The depth of pessimism about prospects for the eventual recovery is evident in the rejection, by 54% to 39%, of the view that "in the long run, younger people will enjoy a more prosperous life than their parents".

A gender gap in voting intentions is again a feature of the monthly poll. Caution is required in reading too much into sub-samples of the survey but, before ICM's final adjustments are applied to the headline figures, Labour leads the Tories by eight points among men, 39%-31%, but by 19 points (45%-26%) among women.

Among the minor parties, support was: Green 2%, British National party 1%, Plaid Cymru 2%, Scottish National party 3%.

ICM Research interviewed a random sample of 1002 adults aged 18+ by telephone on 8-10 March 2013. Interviews were conducted across the country and the results have been weighted to the profile of all adults. ICM is a member of the British Polling Council and abides by its rules.