"I look for the hen."
Translation:Hledám tu slepici.
I must agree with va-diim. "Hledám TU slepici" would correspond to "I look for THAT hen" in English. TU is a demonstrative pronoun, same as THAT. I do not think that example from Russian works here. Without specifying the context of the sentence "I look for the hen" and "I look for a hen" should translate as equivalent in Czech.
When you are looking for the hen, you already know which one. The one that always runs away, the one with one eye or the one that is white. If you are looking for a hen, you will be happy with the first bird you come across. Yes, TU is a demonstrative pronoun but Czech does not have articles, so it needs other ways how to express that any bird is not what you are looking for.