Ministers have been forced to intervene and deploy civil servants to shore up a private company struggling to clear a backlog of medical assessments for payments to tens of thousands of terminally ill, sick and disabled people.

In a letter leaked to the Guardian, a senior civil servant says the "one-off" step will be taken because Capita is failing to process the recently introduced personal independence payment (PIP) claims in time.

The benefit, worth between £21 and £134 a week, is meant to cover transport, care and other costs associated with being seriously sick or disabled. Waiting times for assessment have been so long that in some cases people with terminal conditions have died before receiving a penny.

Speaking on the BBC's Andrew Marr Show on Sunday, the work and pensions secretary, Iain Duncan Smith, said his new PIP system would be fairer than the old disability living allowance (DLA). Under DLA, he said, people went years and years without being able to get the right support for their condition. With PIP, Duncan Smith said, many more people would get face-to-face assessments and the right sort of help.

However, the demand for face-to-face evaluations has overwhelmed the company contracted to carry them out – so much so that the DWP may cut back on such assessments to reduce waiting times.

In a letter dated 20 March, DWP staff were told they would be drafted in for a "one-off exercise" to clear Capita's backlog. Work to reduce "the current backlog to a normal level of work … was now moving at pace", the letter said.

It added: "Ministers have also been very clear that they see the need to drive up PIP performance to an acceptable level as a key priority for the department."

In one of three steps to be trialled, the department will cut down on face-to-face referrals by getting its own staff to determine whether people are eligible for PIP using paper-only evidence.

"We are also considering whether DWP case managers could make more decisions earlier in the journey before formal referral" to Capita, the letter said.

The DWP confirmed that a pilot of paper-based decisions was starting in Wales and Bootle, but said it had always stated that if the case laid out in paper submissions was straightforward, not everyone would need a face-to-face interview with a medical professional.

PIP has already been criticised by the National Audit Office, which said in February that a backlog of 92,000 claims had built up and that people were facing "distress and financial difficulties" because of mismanagement by civil servants as well as Atos and Capita.

Last year the coalition disability minister Mike Penning said those diagnosed with illnesses such as terminal cancer would not have to wait longer than seven days to receive extra state aid. Asked why he had not imposed formal targets on Capita and Atos, Penning said he would use his "size 10 boots" to enforce the desired wait times.

Following sustained criticism of their work, last month Atos confirmed it was departing early from another contract similar to PIP. Duncan Smith told the Marr show that Atos would be paying reparations for their failures on the work capability assessment contract.

"We have now actually asked them [Atos] to go, to leave. They didn't ask us to go, we have asked them. And we will not pay a penny for that. They will actually pay reparations for failure to achieve what they were meant to achieve. The taxpayer will not be out of pocket."

Mark Serwotka, head of the Public and Commercial Services union (PCS), which represents frontline DWP staff, said such benefit assessments should not have been outsourced in the first place.

"We have consistently said these sorts of services should be in-house, private companies have no place in providing them and have failed time and again. The DWP needs to invest in staff and resources to ensure disabled people get the support they need and deserve," he said.

Capita said it was dealing with the situation by "taking on extra staff, including healthcare professionals, and extending the hours for carrying out assessments".

It said: "We continue to deliver the assessments in a fair and objective way, providing high-quality reports to allow the Department to make its decision, and giving people the time they need to tell us how their disability affects them."

A DWP spokesperson said: "Ministers are committed to driving up PIP performance and we're in regular contact with our providers to do this, as you would expect.

"We've always said that where claimants have enough evidence they won't need to be seen in person. However, the majority of people will continue to have face-to-face assessments under the new benefit this government introduced, unlike the old system of disability living allowance where only around 6% were seen."