Can the surplus amount paid by the woman from the Zakat money shared between her and her children be considered an advance payment for next year's Zakat?

It is not permissible to count the Zakat you paid for the coming year; rather, it is the Zakat due on her wealth for this year. This is because the Zakat paid on her behalf by another person is not sufficient, as she did not authorize its payment. One of the conditions for the validity of Zakat is the intention of its owner when paying it to the beneficiaries or the agent, as the Prophet, peace and blessings be upon him, said: "Indeed, actions are by intentions, and indeed every person will have what they intended." Therefore, what is sufficient is what she herself paid, not what was paid on her behalf without her knowledge.