What does Claudius pray for just before Hamlet's first attempt to kill him in Act III, iii?
The question you ask refers to Claudius' famous soliloquy, in which he tries to say his bedtime prayers before going to sleep. Claudius is feeling guilty about murdering his brother, a feeling which has been compounded by seeing the "play within a play" that Hamlet had staged earlier that evening. As a result of this guilt, he attempts to pray, but never truly accomplishes the task.
Instead, Claudius first reflects on how his crime "hath the prima eldest curse upon't"-- a Biblical reference to the story of "Cain and Abel." He questions whether prayer can actually help him, asking, "Is there not rain enough in the sweet heavens/To wash it [his hand] white as snow?" Praying serves only purposes, Claudius reflects: to stop one from sinning beforehand and to give pardon after one has already sinned.
Claudius then wonders how to word his prayer. "Forgive me my foul murder" would not sound sincere because he has not given up the rewards he reaped (ex. the crown and the queen).
Finally, Claudius forces himself to kneel in one last attempt to pray. Although he does manage to say some words that "fly up," he admits that his thoughts still remain base and crude. Because his words are insincere, he concludes that they will "never to heaven go."