What is the ruling on swearing to abandon a sin for a certain period and then repeating it, and is expiation obligatory with each repetition of the sin or only once?

The expiation (kaffara) depends on the wording of the oath and the intention of the swearer. If the swearer did not intend repeated breaking of the oath, or if the wording or custom does not indicate such, then a single expiation is required, and the oath is dissolved upon the first breach. However, if the swearer intended this, or if the wording or custom indicates it (e.g., someone who swore not to abandon Witr prayer), then the expiation multiplies with the multiplicity of the act. The sin of committing the transgression increases in this case because breaking the oath is impermissible.