"He is deaf in his left ear."
Translation:Na levé ucho neslyší.
The possessive is extremely redundant here, to the point of sounding wrong, because you simply can't be deaf in somebody else's ear other than your own.
Also, it would have to be "Na své levé ucho neslyší". (or "svoje") (not used anyway)
Saying "Na jeho levé ucho neslyší" actually means he can't hear using somebody else's left ear, not his own. This would only be usable in some kind of absurd macabre sci-fi story.
I know that the English sentence is not bothered - English loves possessives. English needs determiners (such as "the") to function. And since you must use a determiner, why not use "his" as a determiner, right? There are already too many definite articles :) It's a "weak" unstressed "his".
I'm trying to explain to you the Czech logic, so whether English is or isn't bothered is irrelevant. Czech does not need determiners, so using one (such as "jeho" or "svůj") is never done for the sake of grammar -these words retain their full meaning ("HIS"), they can't be turned into unstressed helping words.