Is it permissible for a woman whose husband—addicted to alcohol—has given her permission to manage the money gifted to their children by the state, to give charity from this money to the needy, knowing that most families use these grants to cover basic or luxury living expenses?

If the grant is paid to any of those mentioned, and the husband's situation is as the questioner described, then he should not be enabled to access this money for two reasons: assisting him in disobedience, and depriving those who deserve it, namely the children and the wife. As for the wife giving charity from the money or a portion of it, this refers back to the details in the previous fatwa (definition): if the state pays this money to the children through the mother or the father, then it should only be spent on them, and the mother is permitted to take only what she needs from it.