What is the ruling on committing to a specific supplication for each round of the circumambulation (Tawaf) or the ritual of walking between Safa and Marwa (Sa'i)?

Specifying a particular supplication for each circuit during the circumambulation (Tawaf) is an innovation (bid'ah) because it was not reported from the Prophet, peace and blessings be upon him. What was reported is the pronouncement of "Allahu Akbar" (God is the Greatest) when touching the Black Stone, and saying: (Our Lord, give us in this world [that which is] good and in the Hereafter [that which is] good and protect us from the punishment of the Fire) between the Yemeni Corner and the Black Stone. Apart from that, any remembrance of God (dhikr), recitation of Qur'an, and supplication is general and should not be specific to one circuit over another.