Helen Barnard, acting director of the Joseph Rowntree Foundation, said the group were now calling on ministers to provide an "urgent lifeline" to families by increasing the child element of Universal Credit and the child tax credit by £20 per week.
"The coronavirus crisis has shown us that as a compassionate society we want to support each other and protect each other from harm," she said.
"It’s vital that we build on this to ensure that the pressures on families with children in particular are recognised and acted on.
"Families are dealing with high costs with children at home and many simply haven’t got the income they need to weather the storm.
"This is taking a major toll on parents’ mental and physical health and damaging family life during an intense period for everyone."
She added: "Providing an urgent uplift of £20 per week to families with children claiming Universal Credit or child tax credits can keep many from being pulled into poverty, especially where parents have lost work as a result of the pandemic.
"By taking action now, we can ensure that the human suffering of this tragic pandemic is not compounded by rising child poverty, damaging life chances and holding a generation back in the years to come."
Dan Paskins, director of UK impact at Save the Children, said the extra funds were neeeded to ensure the crisis was not "compounded" by rising child poverty.
"Life at the moment is especially hard for families forced to make impossible choices about whether to put food on the table or money in the electricity meter," he said.
"A £20-a-week increase will give families with children the lifeline they need to pull them through these difficult times."
Paskins added: "Every child should have the toys and books they need to learn and play. They shouldn't have to worry because their parents are struggling with low pay, insufficient benefits, fear of losing their job, debt and rising costs."
Responding to the report, Labour's shadow minister for child poverty strategy Kate Green said: "This heart-breaking report demonstrates how hard the Covid-19 crisis is hitting low income families with children, especially those who were already living below the poverty line before the crisis, and those from BAME backgrounds.
"Measures taken by the Government aren’t doing enough to help parents meet the bills."
She added: "Labour is calling for the two-child limit and the benefit cap to be lifted, and for an end to the 5-week wait for Universal Credit, so that parents can afford the basics without going into debt, and to prevent more children being plunged into poverty."