How can the non-fulfillment of an oath be forgiven, as well as swearing by one's children and not fulfilling it, and how can sins be forgiven?

The expiation for an oath that is broken is to feed ten poor people, or clothe them, or free a slave. If one is unable to do all three, then one must fast for three days.

As for swearing by children, it is forbidden and scholars have considered it minor polytheism (shirk asghar), based on the Prophet's (peace be upon him) saying: "Whoever takes an oath, let him swear by Allah or remain silent," and "Whoever swears by other than Allah has disbelieved."

The expiation for swearing by other than Allah is repentance, a firm resolve not to repeat it, and uttering the word of monotheism: "Lā ilāha illā Allāh (There is no god but Allah)."

Expiation for sins committed by a servant is achieved through repentance, seeking forgiveness, and increasing good deeds.