If a worshipper omits a letter from the opening Takbir (Takbirat al-Ihram) and then repeats it, is it obligatory to renew the intention for the prayer along with it, and what is the ruling for someone who forgets the intention to remove minor impurity during ablution (wudu')?

The opening Takbir (Takbirat al-Ihram) is one of the pillars of prayer according to the majority of scholars. If one omits even a single letter from it, their prayer is not valid, and they must restart the prayer with a renewed intention if a long time has passed. As for the intention for ablution (wudu'), it is formed by intending to remove minor ritual impurity, or to perform the obligatory act of ablution, or to permit that which is not permissible without ablution, such as prayer, circumambulation (tawaf), and touching the Qur'an. If the one performing ablution does not intend any of these, then their ablution is invalid.