Is expiation (kaffarah) obligatory for one who lied under oath to cover up a sin they have repented from?

You do not have to offer expiation, because an oath concerning a past event is not binding. The oath that necessitates expiation is one concerning future matters. An oath concerning a past event is either a yamin ghamus (a perjured oath) if the swearer knows he is lying, or laghw (a vain oath) if he is unaware. Neither of these requires expiation. If the oath concerning a past event is false, the swearer incurs the sin of lying, but there is no expiation due from him.