Elisha saying "My father, my father when Elijah was being taken up to heaven is not a prayer at all, but an simply exclamation that asks nothing.
You left out the rest of what Elisha said. It is not
simply an exclamation that asks nothing, for it is repeated later in the same book. It is a cry for help from the designated deliverer of Israel when it appears that deliverer will be taken from Israel. King Joash uses the same cry to the Prophet Elisha as Elisha used with the Prophet Elijah, and then Elisha provides deliverance through the spirit of Elijah (which you correctly stated was the Spirit of God), according to the faith of King Joash. Elisha chastised King Joash for not having more faith by which his prayer would have been answered with more deliverance from the adversaries of Israel.
Now Elisha was fallen sick of his sickness whereof he died. And Joash the king of Israel came down unto him, and wept over his face, and said, O my father, my father, the chariot of Israel, and the horsemen thereof. And Elisha said unto him, Take bow and arrows. And he took unto him bow and arrows. And he said to the king of Israel, Put thine hand upon the bow. And he put his hand upon it: and Elisha put his hands upon the king's hands. And he said, Open the window eastward. And he opened it. Then Elisha said, Shoot. And he shot. And he said, The arrow of the LORD's deliverance, and the arrow of deliverance from Syria: for thou shalt smite the Syrians in Aphek, till thou have consumed them. And he said, Take the arrows. And he took them. And he said unto the king of Israel, Smite upon the ground. And he smote thrice, and stayed. And the man of God was wroth with him, and said, Thou shouldest have smitten five or six times; then hadst thou smitten Syria till thou hadst consumed it: whereas now thou shalt smite Syria but thrice.
Second Kings, Catholic chapter thirteen, Protestant verses fourteen to nineteen,
as authorized, but not authored, by King James
You left out the rest of what Elisha said. It is not simply an exclamation that asks nothing, for it is repeated later in the same book. It is a cry for help from the designated deliverer of Israel when it appears that deliverer will be taken from Israel. King Joash uses the same cry to the Prophet Elisha as Elisha used with the Prophet Elijah, and then Elisha provides deliverance through the spirit of Elijah (which you correctly stated was the Spirit of God), according to the faith of King Joash. Elisha chastised King Joash for not having more faith by which his prayer would have been answered with more deliverance from the adversaries of Israel. Try as you must, there is no request or need for one by Elisha in "O my father, my father, the chariot of Israel, and the horsemen thereof." Elisha had no need to request anything since he was already promised what he had requested, but expresses grief at the loss of his spiritual father. Neither his exclamation of "My father, my father..." by him nor by Joash was asking anything, but are both are lamentations of honor such as Jews might give at the passing of someone.
Joash weeps over the impending loss of Elisha, who is neither is given nor answers any request, nor even shows sympathy, but in response he continues to fulfill his role as a prophet by foretelling what God will do, yet reproving the king for his lack of zeal.
That you get prayers out of these honorifics expressed due the loss of someone again testifies to Catholic desperation in their private interpretation. Again, give it up and admit it call comes from extraBiblical tradition.