The Bishop of Gloucester, Rachel Treweek has called on the government to increase benefit levels in line with the cost of living each year.
The Trussell Trust said its network distributed more than 1.3 million three-day food supplies in the year to March, a 13% increase on the previous 12 months.
Almost half a million went to children, said the trust, which runs more than 400 food banks across the UK.
A growing proportion of referrals to food banks were due to benefit levels not covering the cost of essential goods and services.
Low income accounted for almost a fifth of referrals, compared with 26% the previous year, followed by benefit delays or changes and debt.
Emma Revie, chief executive of the Trussell Trust, said: "As a nation we expect no one should be left hungry or destitute.
"Illness, disability, family breakdown or the loss of a job could happen to any of us and we owe it to each other to make sure sufficient financial support is in place when we need it most.
A record number of emergency supplies were handed out by food banks in the past year, research has revealed.
"It's hard to break free from hunger if there isn't enough money coming in to cover the rising cost of absolute essentials like food and housing.
"Universal Credit is the future of our benefits system. It's vital we get it right and ensure levels of payment keep pace with the rising cost of essentials, particularly for groups of people we know are already more likely to need a food bank - disabled people, those dealing with an illness, families with children and single parents."
A Department for Work and Pensions spokesman said: "The reasons why people use food banks are complex, so it's wrong to link a rise to any one cause.
"This research is based on anecdotal evidence from a small, self-selecting sample of less than 0.04% of current Universal Credit claimants, whereas Universal Credit is working for the vast majority who claim it.
"It was also carried out before our significant improvements to Universal Credit came into effect at the Budget; such as 100% advances, which support people before their first payment, removing the seven waiting days and two weeks' extra housing support for claimants moving onto Universal Credit.
The Bishop of Gloucester, Rachel Treweek, said: "The latest figures from the Trussell Trust confirm our worst fears, that foodbank use is rising sharply again.
"We also know current benefit levels are no longer sufficient to cover the cost of basic essentials like food, energy and housing.
"While we applaud the efforts of foodbank volunteers across the country, it is completely unacceptable that people should have to rely on charitable handouts in order to feed their families.
"The benefits freeze was intended to be a temporary response to austerity. This report shows that it is time to relax its grip on families and children who are locked in poverty, by ensuring that benefit levels are increased each year in line with rises in the cost of living."
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