Torbay Council says that despite an increase of £4.4m in its children's social care budget, it is still forecasting an overspend of £4.2m on supporting vulnerable children this year.
A key factor in costs rising is an increase in the number of looked-after children being placed in residential care, which, according to council documents are costing "between £5,000 and £10,000 per week".
The documents, which are being discussed at a council meeting today, also show that the rise in residential care use and spending has happened at a time when the looked-after child population has remained stable at around 360 children.
"However, within this headline figure the mix of placements has seen an increasing number of residential placements," the report states.
"In addition, due to complexity of need, the average cost of residential placements has been increasing.
"There remains a significant risk around any further increases in the number of residential placements."
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The council has identified reducing the need to move children from foster to residential care due to placement breakdowns as a priority in cutting spending.
It also wants to see the length of time it takes to help children return to family-based care reduced.
"Preventing the breakdown of existing foster placements, resulting in a move to residential and the extension of placements due to the inability to return the child to a family-based setting must be a priority for action in order to recover the financial position of Torbay Council," states the council documents.
The council estimates that the residential care bill will increase by £830,000 this year. The extra cost of moving three children from foster care into a residential setting will cost an estimated £359,000, while extending the length of a residential placement will cost an extra £222,000 for two children.
The council is also looking to ensure there are enough local placements, and wants to improve recruitment and retention of children's services staff.
A reliance on agency staff to fill vacant social worker posts is a concern being presented to councillors.
The council's children's services were rated inadequate by Ofsted in 2015 and 2018, and the running of the department was taken over by Plymouth City Council in April 2018.
Its most recent Ofsted monitoring report, published in April, described the recruitment and retention of staff as "a huge challenge".
In February, Ofsted said the children's services department is blighted by a "chaotic way of working" and is failing to ensure it is keeping children safe.
Earlier this year, the Local Government Information Unit's annual State of Local Government Finance Survey said that around a quarter of councils are planning cuts in children's services this year as they struggle to cope with austerity and rising demand.
Supporting children with complex needs was cited as the top reason for children's services coming under financial pressure.
A House of Commons public accounts committee session in February heard how a lack of supply of specialist children's home places is ramping up costs and leading to bidding wars between councils.
A Torbay Council spokesperson said: "Like many local authorities Torbay Council has continued to face unprecedented pressures in relation to our children's social care budget, and much of this relates to the increasing costs of residential care.
"We remain committed to ensuring that as many children as possible are living in family settings and we are focussed on providing good support for our foster carers and recruiting more good foster carers. All our plans aim to ensure that we have the right placements for each of our children in care; this includes making sure that children are not staying longer in residential care than is necessary to meet their needs.
"If residential care is the best placement for a child, despite increasing costs, we continue to ensure that the most appropriate placement is secured and supported based on their care plan and needs."