Six councils in north Wales have protested that their schools will receive just pound;1.6 million of the pound;16m Raise (raising attainment and individual standards in education) fund, announced last week by education, lifelong learning and skills minister Jane Davidson.
Some headteachers also want to see every school receive a share of the cash -most of which (pound;13m) came from Chancellor Gordon Brown's public services windfall in last month's Budget.
But the Assembly government, responding to the complaint from Conwy, Wrexham, Flintshire, Denbighshire, Anglesey and Gwynedd, said: "The Raise funding is not for local authorities.
"It is funding targeted at the schools with the largest number of disadvantaged pupils. This is the Assembly government delivering on a social justice agenda for Wales."
Around pound;14.4m of Raise cash is being distributed to 614 primary, secondary and special schools where at least a fifth of pupils are entitled to free school meals. Payments range from a provisional pound;47,000 to Pounds 221,000 for secondaries, and pound;11,000 to pound;31,000 for primaries.
Secondaries in Gwynedd, Powys, the Vale of Glamorgan, and Monmouthshire will receive nothing.
Heads have been told to spend the cash on their most disadvantaged students, and those most at risk of leaving with no qualifications. They will have to show improvement against agreed outcomes as a result of expenditure on, for example, additional literacy and numeracy lessons, improved home-school links, and out-of-hours activities. And LEAs will receive part of pound;1m for learning support for looked-after children, targeted at Years 10 and 11.
The north Wales authorities believe they would have received an extra Pounds 1.5m if the Raise money had come via the usual routes to LEAs, and accused the Assembly government in Cardiff of "showing a lack of commitment to the north".
Iwan Guy, acting director of the National Association of Headteachers Cymru, said Raise failed to target under-achieving pupils in schools with less than 20 per cent of pupils on free meals.
"The Assembly government has left out a whole raft of pupils who are underachieving," he said.
But Dr Philip Dixon, director of the Association of Teachers and Lecturers Cymru, said the move was long overdue.
"If we spread the money out across the board, those who need it most won't get it. Each school would have such small amounts that it would make no difference.
"Research shows if we spend money in a concentrated way on patches of high deprivation that we can make a very significant difference to these children. That's what's happened in England."