What are the authentic invocations (adhkar) that a pilgrim performing Umrah should recite during circumambulation (tawaf)? And should the first circuit begin with praising Allah, extolling Him, sending blessings upon the Prophet, and then supplicating, while prioritizing religious pleas over worldly ones, and with presence of heart?

The Prophet, peace and blessings be upon him, did not transmit any specific supplications or remembrances for the circumambulation (Tawaf) other than the supplication between the Yemeni Corner and the Black Stone: "Our Lord, grant us good in this world and good in the Hereafter, and protect us from the torment of the Fire," and saying 'Allahu Akbar' (God is Greatest) each time he aligns with the Black Stone.

As for the rest of the circumambulation, the circumambulator has the choice between remembrance (Dhikr), supplication (Du'a), and reciting the Quran. It is recommended to frequently remember Allah the Almighty, and to supplicate for whatever good one desires in this world and the Hereafter, and to remember Allah with any legitimate form of remembrance. It is also recommended to refrain from conversation except for the remembrance of Allah, or reciting the Quran, or enjoining good and forbidding evil, or what is absolutely necessary.