What is the reason for the increased desire to practice masturbation during examination periods, and what is the ruling on the expiation for repeatedly breaking an oath in this case?

What you felt, that desire to practice that habit, was from the whispers of Satan. The duty was not to surrender to his whispers and to occupy yourself with studying. Since the act occurred, you must offer sincere repentance and resolve not to return to it.

Regarding multiple oaths for a single act, only one expiation is required of you, which is feeding ten poor people: one mudd (approximately 750 grams) of food for each poor person, or half a sa' (approximately 1.5 kilograms) of other than wheat, or one mudd of wheat. It is permissible to appoint a trustworthy person to give out the expiation. If you are unable to feed them, then you must fast for three consecutive days as a precautionary measure.