What does the Cyclops ask his father Poseidon to do for him?
After Odysseus and his men blind the Cyclops, they sail away from the island. Just as it seems they are going to get away, Odysseus shouts back to Polyphemus, taunting the monster and telling the monster his real name. In his terrible pride, Odysseus confirms his identity so that Polyphemus can tell people who bested him. It doesn't seem to occur to Odysseus that his identity could ever be used against him, but this is exactly what Polyphemus does. He prays to his father—the god of the sea, Poseidon—saying,
"Here me, thou girder of the land, dark-haired Poseidon! If I am truly thine, and thou art called my father, vouchsafe no coming home to this Odysseus, spoiler of cities, Laertes' son, whose home is Ithaca."
Further, Polyphemus prays that if Odysseus is fated to return to his home in Ithaca that he does so without any of his crew remaining, in a stranger's ship, and that he finds trouble when he gets there. In other words, then, Polyphemus asks Poseidon to exact revenge for the injuries done to him by Odysseus.
Polyphemos' request of Poseidon is central to Odysseus' travels. He first prays that Odysseus never reach home. He then asks that if Poseidon cannot fulfill this wish, that Odysseus come home "late and come a broken man - all his shipmates lost, alone in a stranger's ship - and let him find a world of pain at home! (lines 592-5)" This prophesy is repeated by Tiresius, and at the time Odysseus tells his tale to Alcinous, Odysseus has already lost all his ships and men, and it appears he will return to Ithaca in a Phaecian ship.
The manner in which Polyphemos frames his request is important. He identifies Odysseus by name, by his epithet ("raider of cities"), his patronymic ("Laertes' son") and his home. This is to ensure that Poseidon's curse falls precisely on the right man.
After Odysseus and his men trick and then blind the Cyclops Polyphemus, he prays to his father, Poseidon, the god of the sea and earthquakes, and asks him to put a curse on Odysseus and his men.
Poseidon is angered at Odysseus and does what his son requests, and Odysseus and his men are plagued by Poseidon for the rest of their journey.
For more information about the Odyssey, see the links below:
After being outwitted by the clever Odysseus, then taunted by him from the ship as it sails away from the land of the Cyclopes, Polyphemus appeals to Poseidon, his father, to make certain that if Odysseus does, in fact, make it home, it is after an agonizingly long journey and one in which he loses all of his companions.