Is it permissible for a worshiper - whose memorization of the Qur'an is not firm - to pray Qiyam al-Layl with his wife holding the Mushaf to correct his errors, then they both restart the recitation from the beginning of the Qur'an daily without interruption after completing it, and is this considered an innovation, and is it permissible to consistently recite a hizb from the Qur'an every night?

There is no harm in performing optional night prayers (Qiyam (definition) al-Layl) in congregation with one's wife, whether you pray with a Hizb, a Juz', or less or more. However, it is not advisable to make this a regular practice outside of Ramadan (definition). There is also no harm in the wife holding the Mushaf (definition) (copy of the Qur'an) with the intention of correcting the Imam, although refraining from doing so is preferable to avoid disagreement. It is recommended to consistently review and memorize the Qur'an and to present it to a skilled reciter (shaykh) to correct one's recitation.