Is it permissible to break one’s fast on Mondays and Thursdays vowed for the sake of being blessed with offspring, when there are occasions such as an ‘Aqiqah or a wedding, or must one continue fasting, and is this vow valid?

It is not permissible to break a fast obligated by a vow except for a legitimate Shar'i excuse such as illness, menstruation, or similar reasons. The occasions mentioned are not considered permissible excuses for breaking the fast. Whoever breaks the fast because of them must repent and make up for the days they missed. Fulfilling a vow is obligatory, as Allah the Almighty says: "And let them fulfill their vows." A vow is legally valid despite being originally disliked, as the Prophet (peace be upon him) forbade it, saying: "It does not avert anything; rather, it is used to extract something from the miserly." The prohibition is interpreted as dislike in the case of a vow of recompense, or as a prohibition for someone who believes that a vow changes destiny.