Covid-19 infections will continue to spread through classrooms where high numbers of children are attending schools in the lockdown, experts have warned.

Scientific advice group Independent Sage is calling for the definition of key workers to be narrowed and for increased financial support or furlough to be given to those who cannot work amid a large demand for school places.

The group of scientists, chaired by former chief scientific adviser Sir David King, warns that underprivileged children are being exposed to a “greater risk of infection” due to the high number of pupils who are eligible to attend class.

The warning came after the Government told schools not to limit the number of children of key workers onsite during the national lockdown in England – and it said vulnerable children should be strongly encouraged to attend.

Headteachers have been reporting a high demand for places after students in schools and colleges – except children of key workers and vulnerable pupils – were told to learn remotely until mid-February.

Vulnerable children can include “pupils who may have difficulty engaging with remote education at home” due to a lack of devices or a quiet space to study.

The report from Independent Sage says: “First … this undermines the whole point of school closures making the policy less effective and therefore extending the period of closure.

“Second, it exposes underprivileged children to still greater risk of infection.”

Addressing high demand from key worker parents, it adds: “This is in danger of increasing the number in school to a point where the policy becomes less effective and the ability of teachers to deliver remote learning is undermined.”

Professor Christina Pagel, a mathematician and professor of operational research at University College London, said: “The key worker definition has been broadened, a lot more people now count as key workers and that means a lot more people are going to school now than they were in March.”

A poll by Teacher Tapp, of more than 1,600 primary school teachers, suggests that more than a third (35%) of schools had at least 20% of pupils turn up to school on Wednesday this week.

A similar poll, carried out in the first week of the March lockdown, suggests that only 1% of primary schools had at least a fifth of children come in.

Independent Sage is calling for the creation of a national education task force involving Government, councils, teachers, parents and students to “create a more Covid-secure environment in schools”.

The report also calls for the Government to prioritise immediate provisions of laptops and internet access for every school pupil to ensure remote learning can take place during the tighter restrictions.

Professor Stephen Reicher, a social psychology professor at the University of St Andrews, said: “We recognise that, given the extent to which infections have been allowed to run out of control, it is now necessary to close schools down.

“However, it is entirely unacceptable to shut things down without also acting to make schools safe when they reopen, without providing the resources to ensure that all pupils can study remotely, and without developing plans to address the serious harms caused by closure.”

The Department for Education (DfE) guidance, released on Thursday, said children with at least one parent or carer who was a critical worker could attend class – even if parents were working from home.

Some councils and schools have been seeking to prioritise places for parents who were both key workers, or who worked in emergency services.

Paul Whiteman, general secretary of school leaders’ union NAHT, is calling on the Government to allow “reasonable levels of attendance” to be set by individual schools.

He said: “Of course, schools will be trying to offer the maximum number of places to families, but they have been put in an impossible position. They cannot meet the demand created by Government and reduce social mixing in the way the Prime Minister announced.”

On Monday night Boris Johnson warned schools may act as “vectors for transmission” causing the virus to spread between households, when he announced the closures.

Mr Whiteman is calling on the Government to set out how many pupils on-site is “too many” if the virus is to be suppressed.

He said: “The Government acknowledges that schools do play a role in the transmission of the virus. Therefore, there comes a point when occupancy levels might be so high that they work against the efforts to bring down infection rates in communities, as is the national aim. ”

Mary Bousted, joint general secretary of the National Education Union (NEU), said: “If the Government is serious about having more children at school during this lockdown it should reduce bubbles and groups sizes to minimise transmission risks.

“However, with current staff already fully engaged in providing remote learning to the children at home, reducing bubble sizes will require more staff.”

The Independent Sage report also calls for more to be done to address the mental health needs of students both in class and at home during the lockdown.

Speaking at a briefing on Friday, Professor Reicher said: “It’s essential in the short term that we bring in the extra mental health support, the counsellors, into schools so that we can help our children.

“It is an absolutely essential thing, it is absolutely urgent, and the Government needs to do something about it now.”