Catholics consider Mary "blessed among women," "perfected in grace," the "Queen of Heaven" and saint par excellence. Therefore, since the prayer of a righteous person avails much, Mary's prayer will avail more than any other person's.
Mary is mediatrix and has the power to grant all graces.
Perhaps the petitioners are saying that, at least theoretically, God could choose to mediate all graces through Jesus and then Mary.
Even granting this maximalist position, the gulf between Mary's mediation and Jesus' mediation would be as immeasurable as the infinite gulf between Creator and creation, Source of Grace and channel of grace.
But I will add that in a casual reading of this you would have to say there is a wide gulf between "granting grace" and "Channeling grace". Therefore I think it odd that the writers of the petition did not word it "channel"
Also the phrase "authority to influence" is very ambiguous. My intreprator yesterday could not even define that one, he was puzzled by the statement.
Anyway I was just wanting other Catholic opinions.
Becky