Claudius knows that he has done something nearly unforgivable: he has murdered his brother, just as the first-born son of Adam and Eve, Cain, killed his younger brother, Abel, because God favored Abel over Cain. Claudius, thus, refers to his "offense" as "the primal eldest curse" (3.3.40, 3.3.41). He wants to pray to earn forgiveness for his sin of murder, and he asks, "But, O, what form of prayer / Can serve my turn?" (3.3.54-55). He knows that he cannot be forgiven because he is "still possessed / Of those effects for which [he] did the murder" (3.3.57-58).
In other words, Claudius killed his brother, the old king Hamlet, because his ambition tempted him to take his brother's crown as well as his brother's wife. He is unwilling to part with his position both as the king and as Gertrude's new husband. (For what it's worth, he does seem to love her truly.)
Claudius goes on, speaking to himself, suggesting that he "cannot repent" because he does not want to give up all he has gained as a...
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