Not in this case. If Person A said or inferred that Person C kill B, then I see it. S/he didn't. All Person A did was simply tell Person C of the affair. What Person C did after that, regardless of what Person A "knew" s/he would do doesn't make any difference. A person should be held liable and responsible for their OWN actions, not the actions some other person does.

Your question is 'Should Person A be punished for their actions?'

You state nothing regarding the state of mind of Person A. You do not mention their motives.

Legally, it's clear Person A has zero culpability. Person C made their own choice and took action. I go further than this. It absolutely does not matter one whit if person A is hooping mad or a skilled manipulator and trying with all their might and soul to convince Person C to kill Person B. There is still no culpability. Person C acted! It was their choice alone. Otherwise Person C's human license is removed and they are deemed puppet chattel.

Conscience is the issue. Does Person A accept responsibility for their own part in telling person C and the way in which they behave? If Person A is a reasonable and mature person they will try to present the truth in a way that prevents further loss and pain. They will then continue to engage themselves in the active role they chose in this matter and try to see things through for all parties to the most right and peaceful solution. If Person A is bent and unwilling to behave this way then they should at least admonish themselves internally for starting the mess. Hiding behind delivery of the truth is a fool's paradise. The action of telling the other person did have consequences. Internally, Person A must bear that. Further, if Person A is so immature that they cannot even hold themselves to rights internally, then all parties bear additional responsibility for being around and staying around such a dangerously immature person. Listening to their counsel without knowing that even if the truth is delivered, it will be colored and couched in their own weak anger, fear, or desire, offering nothing of true concern to anyone but themselves.

Person A's motive was desire for the death of person B and belief that revealing the affair to person C would cause this to happen. Again, Person A did not advise or encourage person C to to kill, only revealed the affair in the hopes that doing so would lead to person B's demise. It is not unreasonable for person A to acknowledge internally that their actions contributed to person B's death, or even person D's death. How does their internal acceptance or rejection of their role in person B's death affect their punishment though? Are you saying that person A's punishment should be moral guilt for the deaths of person B and/or person D?

Edit: As to why Person A wanted Person B dead, I intentionally did not mention that. Does it matter why Person A wanted B dead? Does it make Person A any more or less responsible for person B's death?

2nd Edit: I am having difficulty understanding your post series0. From what I can tell, your relevant points are as follows. Please tell me if I misunderstood your meaning on these.

1. Person A has no legal culpability.

2. Even if Person A told person C to kill person B, then Person A should still have no legal culpability for this crime, since they did not actually kill person B themselves.

3. Person A has moral, but not legal, culpability for the death of Person A. Multiple lines expanding upon this guilt.

4. Non-germane stuff about the culpabilities of other parties.

The fact that Person A knew that C would kill B is not important.
The fact that Person A had the intention to get B dead is important. He might not have been sure that C would kill B, but if the intention is there, he is guilty for me.
It would be very difficult to prove, though.

Even if C doesn't kill B, I would consider A guilty of attempted murder, but such a situation probably would never even surface.

Ever heard of conspiracy to commit murder? Happens all the time.

As to punishment - The more mature A is the more their guilt will plague them properly, the more immature they are the more it will continue to twist them that way. Living that life is its own punishment although it also affects others. So internally, maybe, punishment is handled.

Externally, from society's viewpoint, we are blind to the internal world of each person for the most part. Therefore we anneal our wounds or fears by exacting further punishment on those we deem criminals. Personally, beyond eliciting their own internal shame to do its work, I feel this does more harm than good. So, to be clear, I would say punishment is best accomplished by group efforts at revealing what has been done so that shame is applied and others are forewarned of a breach in good character. This does not mean relegating Person A to any sort of forever shamed status, I suppose rather like the Scarlet Letter. It must be assumed that people can work through their guilt and integrate their shame in order to grow and become a better person. of course, this is only a theory ...

It happens all the time when there are clear actions, like handing someone a gun or encouraging someone to commit murder. In this case, there is no action that links directly to murder, unless A explicitly voiced this intend or wrote it down.

I would posit that the individuals for whom this is 'punishment enough' are not those who tend to go around comitting murder (mansloughter, etc., may be another matter), or even setting events in motion with the intent that they result in murder.

Most people who commit murder don't "go arround" committing murder. I think teh same can be presumed for those who might set the wheels in motion to commit murder. I suspect that most people who commit one murder, or cause one murder, will likely feel guilt over it.