Is it permissible for a student to prepare herself for a lecture she did not attend, with the supervisor's approval or at her request?

You are not permitted to prepare a lecture that you missed, because that is a lie. The supervisor is also not permitted to allow it, as it involves aiding in sin and betraying a trust. Allah, the Almighty, says: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression. And fear Allah; indeed, Allah is severe in penalty).