Yes, a single mother with a child, but no income, can file a tax return.
There is, however, no reason to file a return if you have have no income.
As @DoninGA states above:
If you have no income of any kind to report on a tax return, then there is no need or reason to file a tax return, with or without a dependent child. You are not eligible for any kind of tax credit if you do not have any earned income. And you cannot receive a tax refund if you did not have any income where income taxes would have been withheld.
