Holyrood was handed control of crisis grants and money to avoid people having to go into care and was given £23.8m to run the scheme by the Department of Work and Pensions. Scottish ministers then allocated an extra £9.2m to the scheme, giving the Scottish Welfare Fund a budget of £33m.
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This prompted Opposition criticism that only £29m was actually disbursed in the first full year of the scheme.
Martin Crewe, director of the charity Barnardo's Scotland, said while more of the money set aside was reaching those in need, it was "concerning that these figures still show that many millions are going unspent".
The Scottish Child Poverty Action Group said: "It's great to see that many local authorities are doing a better job of ensuring that the Scottish Welfare Fund is finding its way into the pockets of hard-pressed households.
"However, these statistics still show big variations from one local authority to the next and it is essential those areas still seeing a significant underspend consider what they can do to raise awareness of the fund, speed up decision-making and make sure grants are as accessible as possible to families, wherever they happen to live."
Jackie Baillie, Labour's welfare spokeswoman, said the figures provided "further evidence that this Government is completely inept at getting resources to the people who need them most."
A Scottish Government spokeswoman said: "The Scottish Welfare Fund has helped more than 80,000 households."
The £4m would be rolled over to next year.