Is expiation (kaffarah) obligatory for someone who broke their covenant with Allah and then repented, knowing that they are 13 years old?

Expiation is incumbent upon an adult. Adulthood is reached upon turning fifteen years of age, or by ejaculating, or by the growth of coarse hair around the private parts.

If you were not an adult, then no expiation is due from you. And if you were an adult but did not utter the words of a covenant, then no expiation is due from you.

However, if you did utter the words of a covenant with Allah and then broke it, then you owe an expiation for breaking an oath, which is: feeding ten needy persons. If you are unable to do that, then you must fast for three consecutive days.