What is the authenticity of the hadith that mentions a man came to the Prophet, peace and blessings be upon him, and informed him that his wife had committed adultery, so the Prophet asked him to reconsider three times, and on the fourth time, he commanded him to go to Umar to narrate the matter to him, and when he did so, Umar struck his neck?

The aforementioned hadith in the question is not found in the books of Sunnah, and it is most likely fabricated. It appears that whoever wrote it mixed it with another hadith of weak isnad (chain of narration), which is the story of two men disputing and referring their case to the Prophet, peace be upon him, then going to Umar ibn al-Khattab, may Allah be pleased with him, whereupon one of them was killed for rejecting the Prophet’s ruling. Scholars have deemed this hadith weak, but its meaning is acceptable in that whoever rejects the Prophet’s ruling is considered an apostate, and their punishment is execution. As for the narration mentioned in the question, it does not permit the killing of the man.