What is Lady Macbeth's "prayer" to the spirits after she learns Duncan is missing?
First of all, concerning your question about Shakespeare's Macbeth, Duncan isn't missing. In fact, he's on his way to the Macbeths' castle, and Lady Macbeth is extremely happy and excited about it.
She thinks of the idea of assassinating King Duncan earlier in Act 1.5 after she reads a letter from her husband. The letter informs her about the prediction that Macbeth will be king. She expresses at that time that she is worried her husband has too many scruples to assassinate Duncan. He has plenty of determination and drive (he's a successful captain, after all), but he may not want to do the actual assassination, though he would certainly be happy once it was over.
In the speech you ask about, Lady Macbeth is "praying" to have every ounce of any feminine characteristics eliminated from her mind. She doesn't want any kindness or mercy to stop her from assassinating Duncan.
She pleads for the "spirits" to "unsex me here": remove female characteristics and make her more like an aggressive, merciless man. She pleads to be filled with "direst cruelty." She pleads for the spirits to replace her mother's milk with poison. Then she pleads for smoke to cover the area so heaven won't see what she is doing and stop her.
Ironically, the fact that she feels the need to undergo this transformation suggests that she is worried not just about Macbeth not being ruthless enough to go ahead with an assassination, but that she is also worried about herself not being able to go through with an assassination. And we find out in Act 2.2 that, indeed, she is not able. She cannot bring herself to do the actual stabbing, because the sleeping Duncan reminds her of her father. There are multiple personality traits in the character of Lady Macbeth.
I assume that you are talking about what Lady Macbeth says in Act I, Scene 5. There, she asks the spirits to "unsex" her. In other words, she is asking to be made into a man. She wants this because she thinks that if she were a man she would be able to act in a much more ruthless fashion.
She knows that her husband wants to be king by this point. But she does not really trust that he has the determination and drive to do what is necessary to accomplish that. So she asks the spirits to unsex her so she can do it.