No 10 is facing fresh questions over its coronavirus testing plan, as it emerged that the government is hoping to exit the lockdown through controversial “immunity passports” and antibody tests that are still not proven to work.

In the face of intense criticism, the health secretary, Matt Hancock, on Thursday admitted for the first time that mistakes had been made. “There will be criticisms made, and some of them will be justified,” he told the daily press briefing.

Hancock revealed that certificates to prove someone is immune to the virus could allow some of the population to go back to work, as he made a new pledge to complete 100,000 tests a day in England by the end of the month.

He attempted to relaunch the government’s strategy while Boris Johnson remains unwell in self-isolation, after a week of pressure about why so few tests are being carried out, especially on NHS workers.

Hancock promised that tests would be expanded from hospital patients and medics to more NHS staff, key workers and finally more people in the community. These will be a combination of tests for live cases of the virus and antibody tests to determine whether someone has previously been infected. Germany is carrying out around 70,000 tests a day, all for live cases of Covid-19.

However, the government was also forced to acknowledge it was not likely to have the capacity to embark on a programme of mass testing for live cases in the general public, as advocated by the World Health Organization and public experts.

Instead, No 10 and health department sources confirmed the general public would primarily have to rely on the potential for an antibody test – but these are “ideally” done 28 days after an infection, to give the clearest indication of whether someone has already had the virus, according to Prof John Newton, a senior Public Health England official.

Newton said the idea of testing all those that have symptoms in the country was “unrealistic” and the as yet unproven antibody test was more likely to be used by people at home.

Hancock said work was progressing with the antibody tests but that the results of many of the early tests had been “poor” and he was more hopeful about later ones that have been acquired. The government has bought options on up to 17.5m antibody tests of different types that are currently being tested, but cannot say definitively when they will be safe to use.

When the antibody tests are available, Hancock said, the government was looking at the possibility of issuing immunity certificates, so that some of the population “can get back to work as much as possible”.

However, some critics fear that this could lead to resentment in the population who have not had the virus, and that people might even deliberately try to get infected in order to obtain an immunity certificate.

Paul Hunter, professor in medicine at the University of East Anglia, told the Guardian that immunity certificates for frontline medical workers would be crucial for allowing staff to return to work and allowing personal protective equipment to be rationed in the safest way.

However, in the wider population, fraud could be an issue, which could rule out home-based testing, and there were concerns about unintended consequences. “People going out to deliberately get infected so they could get back to work is a concern and I don’t know how you’d avoid that,” he said. “Those are big issues.”

After a week spent in isolation with coronavirus himself, Hancock was forced to defend the government on numerous fronts. He told the press briefing:

Johnson’s ultimate target of 250,000 tests a day – first made on March 18 - was still valid.

Premiership footballers should take a pay cut in solidarity with NHS workers who are putting their lives on the line.

Shortages of swabs had been rectified, but the UK was “still tackling the reagents issue, which is a global challenge”.

The UK was having to build its capacity from a “lower base” than the likes of Germany, which is testing around 70,000 people per day.

More commercial laboratories, including universities and private businesses, would be used to accelerate testing after criticism that this “Dunkirk” approach had initially been ignored.

The NHS will have £13.4bn of debt written off, as previously confirmed.

Tensions have bubbled in Whitehall over who is to blame for low testing numbers in the UK, with some political sources blaming Public Health England (PHE) for not placing enough emphasis on the issue in recent weeks.

However, PHE rejected criticism that it had been trying to do all the testing itself and was too inflexible in which chemicals were allowed to be used, suggesting that it was the responsibility of ministers to find private capacity for more tests.

Prof Paul Cosford, emeritus medical director of the public health body, said PHE’s role was to “make sure our labs are doing what they need to do” in terms of testing hospital patients with a clinical need, with NHS staff a second priority.

Conservative backbenchers voiced concern in private about the government’s failure to roll out testing faster, saying there was anxiety that the public mood could turn against the government if it appeared there was no end to the lockdown as a result.

A former supporter Johnson in the leadership contest said they were worried sentiment could turn rapidly: “I think the government could get blown away if people are still inside after Easter and there is no progress on testing.”

Concerns were voiced that Johnson and other ministers had appeared to allow PHE to lead on testing, without asking the independent researchers and the private sector to get involved until Hancock’s announcement. “The fact that there’s not mass testing now is inexcusable,” another Conservative MP complained.

One senior MP said the party would be watching the polls closely and that he was “frankly amazed” that Johnson’s popularity was holding up. A bit like the virus, there might be a two-week lag before the public comes to fully realise that the government is failing to get the testing issue under control, they said.

“There are some in my party who are trying to blame PHE, but ultimately, it will be the politicians who get the blame if the economy collapses because we can’t test NHS workers.

“The death toll will become totemic. If we get thousands of people dying every day for several days, who knows where this will go. It is frightening, and the prime minister looks like he doesn’t know what to do.”

Jeremy Corbyn, the outgoing Labour leader, said: “The fact that we are not yet even testing 10,000 people a day is very, very serious indeed. There are almost half a million [frontline] people working in the NHS and the care sector. Even they have not yet been tested. It is ludicrous. We have got to get on top of testing.”