Its report – Show Some Warmth – finds that four in 10 children in these families say they have trouble sleeping because their bedroom is too cold, while more than half of the parents are suffering from stress, anxiety or depression.
The survey involved 2,500 families, of which 269 had faced energy debts from one of the so-called "big six" energy firms. The children interviewed for the report were aged 10 to 17.
Overall, its findings show that this winter, as many as two million families – with 3.8 million children – are struggling to pay their energy bills.
At some point, 1.3 million of these families (with 2.2 million children) have fallen behind on their bills. And currently in debt are 470,000 families, with an estimated 800,000 children. Children in these families are "significantly more likely to become ill in winter" because of their living conditions, the report states.
Energy companies are legally required to make sure they assess how much families in these situations can realistically afford to repay. They are also required to make it easy for customers to raise concerns.
However, the report claims that "far too often" this is not happening. It says that four in 10 of the families with dependent children who have faced energy debt say they feel intimidated by the energy companies.
Around 500,000 of these families also report being given no choice at all about their monthly repayments.
The report has been published as part of the charity's Debt Trap campaign, which is seeking to raise awareness of the damaging effect that all forms of debt is having on children today.
Government figures show that 3.7 million children are living in poverty today, a figure that is expected to rise significantly. The Institute for Fiscal Studies has estimated that 750,000 more children will be in poverty by 2020. The government says that six in 10 children living in poverty are in low-income working families.
The report calls on the government to change the law so that families with children are treated as vulnerable customers. It also wants the government to ensure that families that have been placed on pre-payment meters because they are in debt do not pay more for their energy as a result.
Furthermore, energy companies should negotiate affordable debt repayment plans, including lowering or suspending debt repayments over the winter, when children's health is most at risk, the report states.
Matthew Reed, chief executive of the Children's Society, said: "Huge numbers of families are struggling to make ends meet, yet energy companies are making them the victims of their damaging debt practices. They urgently need to show some warmth and stop failing vulnerable children.
"Energy companies across the board need to follow their agreements to work with, not against, families in energy debt and find decent solutions. No child should be made to grow up in a cold home because their parents are trapped in an energy debt trap."
For more on the Debt Trap campaign, visit www.childrenssociety.org.uk/the-debt-trap-end-the-damage-to-children