Delivering the 2018 Budget today, Hammond said the money will be provided over the next five years to address the social care pressures faced by councils. However concerns have been raised that the money provided in the budget is inadequate, and there has been criticism that nothing has been set aside for the early years sector.
Hammond said departmental spending allocations will be settled at the Spending Review next year - however a small number of areas will be provided further support now "in order to deliver necessary certainty for forward planning". In addition to £85m for children's social care, councils have been promised a further £650m for adult social care.
"Local government has made a significant contribution to repairing the public finances, and this budget ensures local councils have more resources to deliver high-quality public services," Hammond said.
"I recognise the immediate pressure local authorities face in respect of social care.
"Building on £240m for social care winter pressures, announced earlier this month, we will provide a further £650m of grant funding for English authorities for 2019/20, and an additional £45m for the disabled facilities grant in England in 2018/19.
"And we will invest a further £84m over the next five years to expand our successful children's social care programmes to 20 further councils with high or rising numbers of children in care, allowing councils to improve services for older people, for people with disabilities and for children in care now, while longer term funding decisions will be made at the spending review."
Department for Education figures show that the number of children in care is rising at its fastest rate in five years. There were 72,670 children in care in the 12 months to the end of March 2017, compared with 70,440 the year before and 69,480 in 2015.
Action for Children said the extra £84m is a welcome acknowledgement of the pressures councils are under, but will not bridge an estimated £3bn funding gap children's social care is set to face by 2025.
"The government must commit to ongoing financial support for cash-strapped councils so they can provide the children's services that vulnerable families and children urgently need," the charity's director of policy and campaigns, Imran Hussain, said.
Adam Lent, director of the New Local Government Network think-tank, said the amounts being provided are not sufficient to address the issues councils face.
"Local public services have endured more cuts than any other part of the public sector over the last eight years," he said.
"This has thrust councils into a growing financial crisis. The Chancellor's long list of one off and relatively small cash boosts are welcome but are really nothing more than sticking plasters.
"If austerity is genuinely to end then councils need a long-term settlement that delivers financial sustainability. Councils will now look to the Spending Review next year for this but given the Chancellor's modest prediction for spending growth, they will look forward more in hope than expectation."
Hammond also announced that the forthcoming NHS 10-year plan will include a new mental health crisis service with comprehensive mental health support available in every major accident and emergency department.
This will include the creation of children and young people's "crisis teams" in every part of the country, as well as more mental health ambulances, more safe havens in the community and a 24-hour mental health crisis hotline.
"These new services will ensure that people suffering from a crisis, young or old, can get the help they need, ending the stigma that has forced too many to suffer in silence and ending too the tragedy of too many lives lost to suicide," he said.
Last week, figures published by Ofsted, revealed that the number of suicides among children who are either in care or are known to social services as being at risk of abuse or neglect increased by more than 30 per cent in the last year.
Ali Oliver, chief executive of the Youth Sport Trust, said: "We are pleased to see the Chancellor of the Exchequer pledging support to address the decline of children's mental health and tightening school budgets in today's budget, but I really don't think we have yet gone far enough to reverse the current decline in this area and the overall wellbeing of this generation.
"We know from the evidence base that young people today are struggling to survive, let alone thrive in the complex context they live within. We hear frequently from schools how declining mental health is a serious barrier to educational achievement and so this is not only a health issue.
"It is our belief that these issues demand proactive, preventative approaches to mental health rather than just increasing investment in addressing the symptoms when they arise."
Kevin Courtney, joint general secretary of the National Education Union, said: "The Chancellor has shown in this budget the depth of his ignorance on school funding. Schools have a £2 billion shortfall in funding a year - which is set to get worse. Capital funding has been cut by a third. A £400m one-off payment for ‘little extras' will do nothing to address this.
‘The government has promised more money for potholes than schools in this Budget. Schools are struggling to provide a full and well-rounded education and many schools have fallen into debt - money for ‘little extras' won't cut it. Parents, teachers, headteachers and school staff will be dismayed."
Meanwhile, Neil Leitch, chief executive of the Pre-school Learning Alliance, said it is "incredibly frustrating" that the government has not pledged more funding to ensure the long-term sustainability of the early years sector in England.
"Survey after survey and study after study has shown that there is simply not enough funding in the childcare system," he said.
"The government's own commissioned evaluation of the 30 hour offer found that the scheme is not only having a negative effect on many providers' finances, it's also resulting in higher fees and charges for a significant proportion of parents.
"How much more evidence does the government need before it finally commits to addressing this issue? How many more pre-schools, nurseries and childminders need to close their doors before someone at the DfE has the courage to say: 'We need to look at this again'?"