Is repeatedly supplicating for one's deceased mother in the second prostration of every rak'ah and in every prayer, while supplicating for oneself in the first prostration, considered an innovation (bid'ah)? And is there a way to compensate for the prayers that one's mother missed due to illness and incapacity?

Supplicating for the deceased is one of the best things that can benefit them, especially if the supplication comes from righteous children. Prostration (sujood) is among the times when supplications are answered. The questioner may supplicate for himself for rectitude in his affairs and ease in obtaining lawful sustenance, and he may supplicate for his mother for forgiveness and mercy in all prostrations or in some of them. However, one should not dedicate a single prostration solely for supplicating for her. Regarding his mother's missed prayers due to illness and inability: if she was in a state of unconsciousness, there is nothing upon her. If she was conscious and it was obligatory for her to pray in the manner she was able or to make up the prayers, it is not legislated for him to make them up on her behalf after her death. Rather, he should frequently supplicate for her, seek forgiveness for her, and give charity on her behalf.