Is an oath sworn on the Qur’an – as a condition for financial aid – to refrain from approaching women or committing adultery, considered a yamin ghamus (a perjurious oath)? And upon breaking it, is expiation (kaffara) required, or repentance? And what is the difference between a yamin ghamus and an oath that requires expiation?

What you said is a binding oath, which, upon breaking, necessitates an expiation (kaffarah) of an oath: feeding ten poor persons, or clothing them, or emancipating a slave. If you cannot find these, then fasting for three days.

As for the Ghamus Oath, it is swearing falsely while knowing it to be untrue. The majority of scholars hold that it has no expiation due to the gravity of its sin; rather, repentance and restoring rights are obligatory. Imam Al-Shafi'i and one narration from Imam Ahmad obligate expiation for it.