Is expiation required of one who swore, "I swear by God that I will do what is in the paper," then added a clause to the paper exempting his days of illness during which he was unable to commit to what was in it?

If you swore by Allah to complete what you wrote on the paper within specific days, and you did not make an exception for illness, then if you become unable to complete it due to illness or otherwise, you have broken your oath and expiation is required of you. However, if you did not specify a particular time for completion, you do not break your oath unless you despair of doing it. What you added to the paper separately from the oath has no اعتبار (legal weight), because an exception is conditioned upon being directly connected to the oath.