What is the ruling on one who broke an oath made on the Qur'an to abandon masturbation during Ramadan, and is it permissible for him to pray for illness upon himself to expiate the sin?

If you did not utter the oath, then there is no expiation upon you. But if you uttered it and then broke it, then expiation is due for each oath. The more cautious opinion is that of the majority, which states that the expiation should be multiplied. Your supplicating against yourself is a mistake and a violation of the Prophet's prohibition (peace be upon him). You must now supplicate to Allah to remove the affliction from you and heal you. If you committed acts that break the fast during the daytime in Ramadan, knowing that they break the fast, then you must make up those days. But if you were ignorant, then no making up is required of you.