In "The Crucible," what does Reverend Hale ask Danforth to do?

There are a number of things reverend Hale asks Judge Danforth to do. The first thing he requests is that the judge listen to the evidence Giles Corey is about to give in defence of his wife, Martha, who has been arrested. Reverend Hale tells the judge:

Excellency, he claims hard evidence for his wife’s defense. I think that in all justice you must -

The arrogant judge interrupts him here and says that Giles should present evidence. In the same breath, he reminds the Reverend of the fact that he is aware of what the procedures are.

Secondly, the reverend also beseeches the judge to give Mary Warren a hearing. She is there to testify about the girls' deceit and Abigail Williams' complicity in having Elizabeth Proctor arrested. Once again, the judge practically dismisses the reverend by raising his hand at him.

Reverend Hale's repeated intervention indicates that he, at this point, is starting to doubt the veracity of the girls' testimony. He wishes the court to listen to what he believes is the truth, but the pompous and supercilious Judge Danforth gives him no chance. Reverend Hale feels helpless and hopeless.

When John Proctor addresses the court to give evidence, Mr Cheever remarks that John ploughs on a Sunday, an apparently un-Christian act. Once again, Reverend Hale requests that Judge Danforth not judge John on such a feeble accusation. The judge, in his response is quite abrupt, saying: 'I judge nothing', which is, ironically, not true. 

Reverend Hale clearly wants the court to use reason and logic and not emotion, in its judgement. He tells Judge Danforth that the court should consider that not everyone accused is necessarily guilty of trying to topple Christ. Once again, the judge rejects his assertion, stating that uncorrupted men will not fear the court, which, once again, is ironic, for there is great fear for the court since all who have been punished thus far are actually innocent!

When John Proctor comes forward to present his deposition, Reverend Hale begs Judge Danforth to send him home to get a lawyer so that he can present his case professionally. At this point it is clear that the Reverend is overwrought with guilt, for he cries out:

Excellency, I have signed seventy-two death warrants; I am a minister of the Lord, and I dare not take a life without there be a proof so immaculate no slightest qualm of conscience may doubt it.

I have this morning signed away the soul of Rebecca Nurse, Your Honor. I’ll not conceal it, my hand shakes yet as with a wound! I pray you, sir, this argument let lawyers present to you.

The judge deems this a personal affront and once again, dismisses the Reverend's request using a clever argument.

When Mary Warren accuses John Proctor after the girls once again put on a marvellous act of being attacked by the witches' spirits, the reverend again and again tries to intervene, but is dismissed with contempt. He is finally so disgusted that he denounces the proceedings and storms out of court.

Finally, when John Proctor has confessed, it is Reverend Hale, on his return, who asks the Judge that he postpone the hangings for a week and put up a notice in town that he seeks the villagers' confessions to show his mercy. However, the judge insists that he will continue for he, 'cannot withhold from them the perfection of their punishment'. Clearly, the judge is not going to be told what to do.

Reverend Hale's final request to the judge is that he lets John Proctor sign his confession, instead of asking him to implicate others as witches. This, in the end, is the only request that the judge does grant.