The charity's 2018 annual report says 4.1 million under-18s are living in households with insufficient money.
It adds that relative poverty, measured as 60% of median household income after housing costs, now affects more than one in five people - half of them in families where at least one person is working.
Charity chief executive Campbell Robb said the figures showed a "rising tide" of poverty that is increasingly affecting working families.
He said: "It means more families are trapped in impossible situations, struggling to pay the bills, put food on the table and dealing with the terrible stresses and strains poverty places on family life."
This latest research comes less than three weeks after a highly critical UN report on the impact of austerity in the UK, which accused the government of "callous" spending policies that hit the poorest the hardest.
Both reports have been welcomed by a help centre in Halifax, in a part of Yorkshire with some of the highest levels of disadvantage in the country.
The Gathering Place, a partly council-funded food, advice and accommodation centre run by the Calderdale Churches Together charity, said demand for its services is rising and it is extending opening hours to cope.
Support worker Chris Wright said problems with the roll out of Universal Credit, the government's single payment system to replace a range of benefits, was one of many factors forcing more people to rely on the centre's services.
He told Sky News: "It's across the board, people who've been in work, who for whatever reason they're not in work any more, all backgrounds, all walks of life, we're seeing everybody."
The centre provides 80 hot meals a week and 100 emergency food parcels, and from next year will keep its dormitory for homeless people open for six months of the winter instead of four as in previous years.
Paul Maffia, the centre's caretaker who was once homeless himself, wants a fairer society and blames "MPs and bankers" for not understanding how policies affect the poor.
"The bedroom tax and universal credit, they don't know it, they don't know how it affects you," he said.
The government has dismissed the report, countering it with a different calculation of "absolute poverty" based on 2011 incomes adjusted for inflation.
A spokesman for the Department for Work and Pensions said: "We disagree with this report, and there are now one million fewer people living in absolute poverty since 2010, including 300,000 children."
The official criticism of the Joseph Rowntree Foundation report comes two weeks after Work and Pensions Secretary Amber Rudd attacked the UN report into austerity and poverty in the UK over what she described as the "extraordinary political nature" of its language.
By Gerard Tubb, north of England correspondent