Do the children have a right to the compensation their father received after their mother's death, and would the father be sinful if he did not ask for their forgiveness?

If the compensation intended is the blood money for murder, then it is an inheritance for all heirs, to be divided among them according to Islamic law. Men and women will receive their share from it, with the exception of the killer. The children of the deceased woman have a right to her blood money, both males and females, and her husband is not permitted to deprive them of this right. They should request it gently and kindly. If he withholds it from them, he is unjust, and they have the choice between forgiving him or taking the matter to the Sharia court. Forgiveness does not require a request; rather, they can forgive him of their own accord.