The Eye Patch Goon is an unnamed henchman of the penguin, a soldier in the Cobblepot crime family, and a former soldier of the Maroni Crime Family.
He mostly appears in the background of scenes involving Penguin. He is one of the men who is partnered with penguin during his beginning in the Maroni Crime Family. When penguin becomes the king of Gotham, he is one of the guards at his mansion. When penguin goes on the run after the murder of Theo Galavan, The Man, along with many of penguins men, went to work for Butch Gilzean, who took over the criminal underworld in penguins absence. However, penguin soon later on returns as king of Gotham. Even through all of penguins different falls from power, the man still remained working for penguin, all the way to no lands land. It’s unknown what happened to him afterwards, but it’s most likely he’s one of the men who remain loyal to penguin after his release from Blackgate.
judging from his first appearance all the way back in season 1, and his continued work for his boss, he is one of penguins most loyal men.