Someone who murders b’shogeg must stay in an ir miklat until the death of the kohen gadol. The Mishna writes that the mothers of the kohanim would deliver food to the arei miklat to prevent those inside from praying for their son’s demise . What does the kohen gadol have to do with the plight of the murderer? The gemara explains that the kohanim are to blame for the murderer being there, as they should have davened for rachamim.
How does the gemara know that the kohen did not daven? Maybe the kohen did his part and davened, but it's the zechuyos of the murderer and/or the nirtzach that are wanting? If I daven all day for the Mets to win, it could be that they lose because my davening is not good enough, but it could also be that the Mets just don’t have anyone on the team who can hit or field! (Yankee fans, please substitute Yankees in that sentence.)
Secondly, the gemara says that even if the kohen gadol is appointed just before the verdict on the murderer is announced, he is still to blame for not davening that the court rule in the murderer's favor. What good will his prayers do at this point? It’s like a kid in school who takse a test and then davens that he/she will get a good grade. The answers are on the paper already – tefilah is not going to change the facts. Beis din has to pasken based on the events that occurred. Is the kohen supposed to daven for beis din to make an error and rule incorrectly?
The underlying assumption of these questions is that beis din works like a computer – you plug in the facts, and out comes the psak based on some formula. It doesn’t work that way. The Mishna in Makkos writes that R’ Akiva and Rabbi Tarfon boasted that they could completely do away with capital punishment. They would ask detailed questions of the witnesses that they would inevitably be tripped up and have their testimony impugned. Rabban Gamliel responded that this approach would inevitably encourage crime. Surely R' Akiva would not want to free the guilty and surely Rabban Gamliel did not have in mind that the innocent should be handed a death sentence! R' Shaul Yisraeli explained the machlokes here based on Tosfos' view that in addition to the required derishos v’chakiros, beis din has a right to further question witnesses. The witnesses are not bound to answer these additional detailed queries, but if they do, they must answer correctly. There is no machlokes between R’ Akiva and the other Tanaim – it is a matter of what circumstances allow for. Where society was as a whole ethical and crime, especially a heinous crime like murder, was an aberration, beis din could seek to avoid capital punishment knowing that this would not lead to an increase in immoral behavior. But when society was plagued by social and moral ills, a tough stance on crime was needed and beis din could not afford to not mete out capital punishments.
The kohen gadol, as a leader in Klal Yisrael, is responsible for the moral state of society as a whole. That is what he should be davening for and working to improve. Had society been more just and more ethical, it would be possible for beis din to interrogate the witnesses more fully and see that no penalty of ir miklat or capital punishment is ever carried out. But because society is not on that level, the murderer must suffer a more severe penalty for his deeds.
(Side note: isn't R' Akiva's position fascinating? Even though there is a clear mandate of u'bi'arta ha'ra mi'kirbecha and testimony to incriminate the wrongdoer, he uses a loophole of sorts [that's probably too charged a word to use] to circumvent the seemingly inevitable outcome of death because of his belief that justice would not be served in that way. Isn't that allowing a subjective value judgment of what the "right" outcome should be to enter into the equation? Isn't that deliberately tilting the scales one way? Something to ponder.
Side note #2: see Michtav m'Eliyahu vol 3 p. 87 for a different approach to understanding the power of the kohen's tefilah and his responsibility.)
On a completely different note, apropos of the 9 days, the Tiferes Shlomo derech derush comments on the pasuk describing the boundaries of Erertz Yisrael, "V'hisavisem lachem l'gevul keidmah...." (34:10), literally translated, "You will turn the border..." that perhaps the word "v'hisavisem" is an allusion to the word "ta'avah," desire. We have to have a ta'avah for Eretz Yisrael -- to want to be there, to want the Shechinah to return there.