John Dickie, director of the Child Poverty Action Group in Scotland, said ministers should ensure the Scottish Child Payment holds its value in real-terms next year.
The UK Treasury has said the Scottish Government will receive more than £600 million of extra funding as a result of the income tax reduction and stamp duty cut down south.
This is due to adjustments made to the Barnett formula, the system that determines funding for the devolved governments.
The First Minister could use the cash to replicate the tax cuts in Scotland or to help fund other measures.
Ms Sturgeon has already all but ruled out following the UK Government in scrapping the top rate of income tax for those earning £150,000 a year or more.
Mr Dickie told The Scotsman: “Rather than mimicking widely criticised UK tax cuts that mostly benefit the better off, the Scottish Government should continue to do the right thing and prioritise tackling child poverty.
"That means investing any additional resources to ensure the new Scottish child payment holds its soon-to-be £25-a-week value in real terms next year, and making sure the childcare, housing and employment actions that underpin its child poverty strategy are adequately resourced.”
Scottish Labour finance spokesman Daniel Johnson said: “The Tories decision to hand tax cuts to the rich is not only morally bankrupt, but economically illiterate.
“In Scotland we have a chance to take different choices, by making sure this money is targeted and reaches those who really need it, as well as breathing fresh life into our economy.
“To make sure this money is spent right, the SNP need to drop the secrecy and clever accounting, and deliver total transparency around Scotland’s finances.”
Scottish Greens finance spokesman Ross Greer, whose party has a co-operation agreement with the SNP, said: “The UK Government cut Scotland’s budget by more than 5 per cent at the start of this year and then their economic incompetence resulted in a further £1.7 billion loss of value as a result of runaway inflation.
"Even if the Treasury are telling the truth about additional funding coming now, it is only a small fraction of what they’ve just cut.”