There is no specific supplication for each circuit; rather allocating a specific supplication to each circuit is an innovation (bid‘ah), because that was not narrated from the Prophet (blessings and peace of Allah be upon him). The most that was narrated is to say takbeer (Allahu akbar) when touching the Black Stone and to say Rabbana aatina fi’d-dunya hasanah wa fi’l-aakhirati hasanah wa qinaa ‘adhaab an-naar (Our Lord! Give us in this world that which is good and in the Hereafter that which is good, and save us from the torment of the Fire) [al-Baqarah 2:201] between the Yemeni Corner and the Black Stone. As for the rest, it is dhikr in general and Qur’aan and du‘aa’, and there is nothing that is specifically to be recited in one circuit and not another. End quote.
Majmoo‘ Fataawa Ibn ‘Uthaymeen, 22/336, 337
See also the answer to question no. 34644