Has the husband, who swore on the Qur'an not to return to treachery, been absolved of his oath, given that the oath was conditional on his children, and then the wife renounced it out of fear for them, and what should the wife do about that?

The ruling on the oath sworn by your husband does not fall by your statement, "I waive this oath." The ruling of the oath remains valid. If the husband breaks the oath, expiation is incumbent upon him, which is feeding ten poor people. If he cannot find them, then fasting for three days. There is no need to fear for the children from the effect of this oath. It is obligatory for the husband to fear Allah and repent sincerely. The path to repentance is not by swearing an oath to abandon sins, but rather by being truthful with Allah, seeking His help, and striving against the self. Expressing illicit relations as marital infidelity suggests that they are abhorrent only because they are a betrayal of the wife, which is incorrect. Rather, they are an act of disobedience to Allah and a transgression of His limits.