Why did Chief Inspector Kido deliberately amputate his left hand?
He did so after he acceded to work for the Yakuza in The Man in the High Castle S4E10?
Quoting from Wikipedia
Yubitsume ("finger shortening") is a Japanese ritual to atone for offenses to another, a way to be punished or to show sincere apology and remorse to another, by means of amputating portions of one's own little finger. In modern times, it is primarily performed by the yakuza, one of the most prominent Japanese criminal organizations.
Takeshi Kido traded his life for his son to the Yakuza. As he is now under Yakuza he must swear allegiance. Since Kido had disrespected the Yakuza when he was Chief Inspector, to atone for his actions he performs Yubitsume.