What is the expiation for someone who made a covenant with Allah to abandon a certain sin and to pay a specific amount if they committed it, but then fell into that sin and did not fulfill their covenant? Is this covenant binding upon them for life? And what should someone do who has obligated themselves to pay an amount every time they commit a sin?

Abandoning sin is obligatory. So, if a Muslim swears an oath or makes a vow to abandon it, then the abandonment becomes confirmed for him, and it is not permissible for him to break his oath. If he breaks an oath made out of obstinacy or anger (Nathr al-Lajaj wal-Ghadab), he must offer an expiation for breaking an oath (kaffarat yamin (definition)), based on the Prophet, peace and blessings be upon him, who said: "The expiation for a vow is the expiation for an oath." After the expiation, his oath is dissolved.