Is it permissible to pay Zakat to minor children whom their wealthy father and paternal siblings do not support, while their mother works, has a moderate income, and cannot provide for their basic and educational needs?

The financial support of your children is incumbent upon their father, and he must provide them with what is due to them in terms of maintenance. If he is poor or refuses, then it becomes your responsibility. Based on this, it is not permissible for you to give your Zakat money to them. However, if you are unable to provide for them, then there is no harm for someone else to give Zakat to them, nor is there any blame upon them for taking it, as they would then be considered poor.