Children with terminal illnesses, living with physical or mental disabilities, or having other special educational needs are among the most vulnerable pupils there are.

Designated as having high needs, they are supposed to be supported to ensure that they have equal access to education. But providing for children who have the odds stacked against them is anything but straightforward.

The nature of many individual cases - which include children with special educational needs and disabilities (SEND), as well as those in hospital schools or in pupil referral units - makes them complex and expensive to manage.

And now it appears that the education of high-needs pupils, many of whom rely on receiving individual support to enable them to learn, is in serious jeopardy.

Tes has uncovered a financial gulf between the funding being provided by central government this year for high-needs pupils and the amount that local authorities say is needed.

In 2017-18 alone, the shortfall in high-needs funding stands at hundreds of millions of pounds, our analysis shows.

Data from more than half of all local authorities across England, obtained via Freedom of Information requests, shows a projected shortfall this year of £226 million.

Assuming that this pattern is replicated across the rest of England’s 152 local authorities, the figure rises to £409m.

Chris Keates, NASUWT teaching union general secretary, says the union “is increasingly concerned about a growing crisis” in SEND education and adds “the education and wellbeing of some of the most vulnerable children and young people in society is being jeopardised through inadequate resourcing of their needs”.

There needs to be significant extra funding if we are to “avoid a full-blown crisis in society’s treatment of children and young people with SEND,” according to Ms Keates.

An analysis of annual budgets over the past four years shows the funding shortfall has more than tripled since 2014-15, when it amounted to £70m - or £127m, projected across all local authorities.

Since then, councils’ funding gaps have increased exponentially. In some cases, the deficit now equates to more than a third of the funding they receive for high-needs pupils.

Barnsley Council has the largest shortfall on this measure, with £5.1m this year - which equates to 37 per cent of the £13.7m it received for pupils with high needs.

It is closely followed by Kingston Council, which has a funding gap of £6.6m. This represents 35 per cent of its £18.9m high needs funding for 2017-18.

Other councils with large shortfalls equating to at least a quarter of their income are South Gloucestershire, Rotherham, Bury and Reading.

Fiscal pressures are set to continue next year, with a number of local authorities already forecasting significant shortfalls in 2018-19 amounting to up to 30 per cent of their high-needs budget.

The three most commonly cited measures being taken by local authorities to deal with these deficits are: using their reserves, transferring amounts from other funding blocks and carrying forward shortfalls into the next financial year.

But what is behind this funding squeeze? West Sussex headteacher Jules White, from the Worth Less? campaign for fairer funding, says: “Demand for support for those youngsters with SEND and additional needs continues to outstrip resources, staffing and capacity”.

The number of children in England with a statement of SEND or an education, health and care plan has risen from 223,945 in 2010 to 242,185 in 2017 - an 8 per cent increase - according to Cordis Bright, a public services consultancy.

Another factor is thought to be the 2014 Children and Families Act, which extended the age range of pupils with SEND that local authorities are responsible for to 25. But, campaigners say, the extra responsibility was not matched by additional funding.

Some special school headteachers are warning that budgets are now so stretched that pupils are being turned away.

Grahame Robson, head of Manor Green College special school in Crawley, West Sussex, says: “I have already reached the stage where I am having to send students home because we don’t always have enough staff to guarantee their safety”.

Concerns over the scale of the financial crisis facing high-needs provision appear to be escalating. Last month, it emerged that there is a £100m shortfall in the SEND budget across London. And earlier this year, Tes revealed how high-needs pupils will lose out under the new funding system in which councils have less flexibility to cover shortfalls using money from elsewhere in the schools budget.

Heads at mainstream and special schools “have repeatedly described a high-needs system that is at breaking point in their local authorities”, according to Valentine Mulholland, head of policy at the NAHT headteachers’ union.

Local authorities are also facing court decisions that force councils to adopt more expensive care packages. Peter Lowe, head of strategic business and finance at Bury Council, says: “SEND tribunals exacerbate councils’ financial dilemmas by not being aware of the funding implications of their decisions.”

He cites a case where an education health care plan costing £167,000 annually had been prepared for a pupil, but a tribunal said the council must provide enhanced provision costing more than £420,000 for each year the young person is entitled to support. This is expected to amount to around £2 million in total.

A Department for Education spokesperson said: “We are investing an additional £1.3 billion in schools and high-needs funding and have protected this budget in real-terms over the next two years. Under the new high-needs national funding formula, every local authority will see an increase in funding.‎

“Where authorities are finding costs difficult to manage within their allocations of high needs funding, there is continuing local flexibility to ensure that councils can work with the schools in their area and direct funds to where they are most needed.”