All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Supplicating is desirable during Tawaaf but it is not an obligation. Also, it is desirable for a person performing Tawaaf to supplicate with any legislated supplication and there is no difference in this regard between one round and another.
So, it is not an obligation to say a given supplication in every round of the rounds of Tawaaf. Ibn ‘Uthaymeen said: “It is not reported from the Prophet during Tawaaf that he used to say a specific supplication in every round (of the rounds of Tawaaf).”
Furthermore, Ibn Taymiyyah said: “There is no specific mention of Allaah in Tawaaf that was reported from the Prophet ; he did not order this, say this or teach this; rather, he used to say any of the legislated supplications in Tawaaf.”
Allaah Knows best.