Schools are "teetering on the edge" as more than a third of heads reported staff absence levels of over 10% due to Covid on the first day of term.
A snap survey by the NAHT union found the majority (95%) had pupils absent due to Covid, while nearly a third (29%) reported over 10% of children were out of class because of the virus.
The poll of nearly 2,000 school leaders in England revealed staff shortages amid mounting fears over fresh disruption to children's learning.
More than a third (36%) of school leaders said over 10% of staff were absent on the first day of term for Covid-related reasons.
Around a quarter (27%) said more than 10% of teachers were off due to the virus, while nearly one in 10 (9%) said they had more than 20% of their teaching staff absent.
Half (50%) of heads said they were using supply teachers to cover classes - but over a third (37%) were unable to source the supply staff they need.
Education Secretary Nadhim Zahawi has told schools to merge classes if staff shortages grow too high, in order to keep children in school.
However only 4% said they had sent classes or year groups home and fewer than 7% of heads reported having to merge classes or year groups.
NAHT General Secretary Paul Whiteman said: “Schools are doing everything they can to maintain education for pupils in the face of very challenging circumstances.
"Staff absence on the first day of term was considerable and school leaders have been doing what they can to redeploy teachers and other support staff to avoid being forced to combine classes or send groups home.
“Given that this is a snapshot of just the first day of term, this is a very worrying picture. Infection rates – and therefore absence due to illness – could very likely rise as the term progresses, and already half of schools are having to turn to supply staff, with many finding they cannot secure suitable cover.
"Many schools are teetering on the edge and the next few weeks at least will undoubtedly continue to be an incredibly challenging time."
Meanwhile, a separate survey found Government plans to provide 7,000 air purifiers for schools and colleges will fall thousands short.
A poll by the NASUWT teachers union found (56%) of the nearly 2,000 teachers who responded said they do not have access to a CO2 monitor in their classroom.
Even for those that do have a classroom monitor, nearly one in ten (9%) said it was not working properly.
Shadow Education Secretary Bridget Phillipson said: "The government has no plan to prevent nor to manage thousands of staff being off school due to Covid.
“Ventilation, vaccination and testing are key to keeping schools running but the government’s chaotic, mismanagement means tests aren’t available, only half of eligible pupils have been vaccinated, and children are being left to learn in coats next to open windows as temperatures plummet.
“Again and again our children have been treated as an afterthought by Conservative Ministers, seemingly unable to act until it’s too late."
A Department for Education spokesperson said: “We’ve supported schools to continue classroom learning for pupils through encouraging former teachers to step in and extending the Covid workforce fund for schools that are facing the greatest staffing and funding pressures.
“We’ve also asked schools to have contingency plans to maximise attendance and minimise disruption to learning, should they have high rates of staff absence, and are working with the sector to share case studies of flexible learning models to support the development of those plans.”
