Boris Johnson is being warned by cabinet ministers to exempt more people from the “pingdemic”, or face a wave of supermarket, Post Office and restaurant closures.

The Prime Minister has announced an exemption that allows “critical" workers to avoid self-isolation after contact with someone with Covid-19, and instead go into work after a negative test.

But there are growing concerns within his own government that he has not gone far enough, given scores of major industries are unlikely to benefit from the scheme.

One cabinet minister told The Telegraph binmen and other public service jobs should be included in the exemption, saying they understood people’s “frustration” with the rules.

A source close to a second cabinet minister called for Number 10 to go even further, allowing workers in hospitality and supermarkets to be able to avoid staying at home.

“What counts as a critical worker? In my view, logistics would count. In my view supermarkets would count. In my view hospitality would count,” the source said. 

It came after two business ministers told the public self-isolation when pinged was a choice and not imposed by law. Paul Scully, the business minister, said “it’s up to individuals and employers” whether to self-isolate after being alerted by the app when explaining the rules.

Lord Grimstone, the investment minister, called the app an “advisory tool” with no “legal” force in a letter to a large employer which leaked.

There was little sign of the so-called “pingdemic” (when people who are alerted by the NHS Test and Trace App to self-isolate after contact with someone with Covid-19) easing off.

The system is being blamed for causing chaos among businesses who say they are unable to function properly with so many members of staff isolating.

More than one million children in England were off school last week due to coronavirus-related reasons, official figures published on Tuesday revealed.

Proportion of children out of school in England for covid-related reasons

Meanwhile, YouGov polling found that 20 per cent of people who are using the app have turned off their bluetooth or contact tracing, meaning it will not work as designed.

The Government has failed to publish guidance on exactly who would be covered by the scheme or specify how many people could apply, leading to business confusion.

Downing Street scrambled to clarify the Government’s position, distancing itself from Mr Scully’s comments and stressing people should still self-isolate if pinged.

A Number 10 spokesperson said: “Isolation remains the most important action people can take to stop the spread of the virus." 

The spokesman added: “Businesses should be supporting employees to isolate, they should not be encouraging them to break isolation.”

Number of contact tracing alerts sent in England

With Covid-19 cases surging, the number of people who are being alerted by the app as potentially coming into contact with someone with the virus is steeply rising too.

More than half a million people were told to self-isolate last week. But business leaders have warned that figure could reach five million if new Covid-19 cases hit 100,000 a day, as predicted.

If pinged by the NHS Test and Trace app, people have to self-isolate for 10 days, even those who have been double jabbed. No amount of negative Covid-19 tests can lead to release. 

Mr Johnson announced on Monday that a “very small number” of named individuals from sectors that are critical to national infrastructure would be able to apply for the exemption.

Those who qualify can return to work after a negative PCR test. He listed people working at “our hospitals and care homes, the supplies of food, water, electricity and medicines, the running of our trains, the protection of our borders and the defence of our realm”.

But Government insiders said that people working in supermarkets, pubs, restaurants, Post Offices or collecting rubbish would not be approved under the scheme.

There was confusion about how exactly businesses who believed they could qualify should apply, with employers told simply to liaise with the relevant Government Departments.

One minister with an economic brief told The Telegraph that such exemptions when approved would only apply to workers pinged in the future, not those already in self-isolation.

Downing Street has doubled down on its refusal to bring forward the date at which people who have been twice jabbed will be able to take daily tests rather than self-isolate.

That is due to happen on Aug 16, more than three weeks away. Mr Johnson has argued self-isolation must remain in place given the heightened risks of those pinged being infected.

Hopes of the NHS Test and Trace app being reformulated, given inconsistencies such as people being pinged when neighbours get Covid, have also been dashed. 

Why the NHS app pings some phones but not others

Multiple figures inside the Government have told The Telegraph there is no prospect of the app being changed to be less broad in terms of who it alerts before the August 16 rule change.

Mr Johnson, Rishi Sunak, the Chancellor, two health ministers and scores of other senior Government officials are still self-isolating after Sajid Javid, the Health Secretary, got Covid-19. 

Tony Blair, the former prime minister, has led calls in recent days for an immediate dropping of the need to self-isolate if pinged, suggesting daily testing replaces it.

Mr Blair has argued that it makes little sense Mr Johnson, who has had Covid-19, is double jabbed and is testing negative, is isolating at Chequers, unable to return to Number 10. 

Supermarket branches, pubs, restaurant and train lines are among those services which have been forced to temporarily shut due to so many staff being told to self-isolate.