What should I do about repeatedly breaking my covenant with Allah to abandon the forbidden, and is the covenant considered an oath that necessitates expiation? If it is obligatory, is it permissible for me to expiate for some of it by feeding the poor and for the remainder by fasting, given my low income, and how can I justify this to my family without informing them of the matter?

You are obligated to repent to Allah for the forbidden act. The aforementioned vow is considered an oath, and an expiation for breaking an oath (kaffarat yamin) is incumbent upon you. If the breaking of the oath is repeated before expiating the first oath, and the object of the oath was one and the same, then a single expiation for breaking an oath will suffice. However, if the breaking of the oath occurs after expiating the previous oath, then another expiation becomes obligatory. If you do not have sufficient funds for expiation, then fast for the remaining oaths. You are not required to inform your family about the true nature of the expiation; you can use allusive language (tawriyah) by saying that you have sought closeness to Allah by feeding the needy and fasting.