"The woman eats that cheese."
Translation:Kadın o peyniri yer.
No, it's not. In this sentence "o" means "that" indicating "that cheese-o peyniri". Just pointing out that the woman eats that cheese, not this cheese kind of thing. Does this help?
"Kadın şu peyniri yer." is one of the alternative translations and it should be accepted as a correct answer as we included it.