Is expiation obligatory for someone who vowed to fast for a week or a month if his beloved returned to him, then he only fasted for two days, and his relationship with her was completely severed?

Your vow to fast if that girl returns to you is a vow contingent upon the occurrence of a sin. This vow is not binding, and nothing is required of you concerning it. Ibn Hajar Al-Haytami mentioned that a vow contingent on the arrival of a person for an corrupt purpose, such as a non-mahram woman, is not binding, similar to a vow of sin. He also stated that Shaykh Zakariyya erred in disagreeing with this opinion. Strive to achieve and maintain your repentance from the forbidden relationship.