Learn Turkish: Yine hata yaptım. - I made a mistake again.

In English we say "a mistake", but in Turkish, you don't always need bir for general, non-specific objects. The phrase hata yapmak works together almost like a single verb meaning "to mistake-make". You could say Yine bir hata yaptım, but leaving bir out is very natural and common.

Why doesn't hata have an accusative case ending (like hatayı)?

The accusative case is only used when the object is specific or definite (meaning "the"). Because you are talking about making "a" general mistake, not "the" specific mistake we just talked about, hata stays in its plain dictionary form.

Could I use a different verb to mean "made"?

No, yapmak is exactly the right verb here. It translates to both "to do" and "to make" in English. The combination hata yapmak (to make a mistake) is a fixed pairing in Turkish, just like it is in English.

Can I put yine at the end of the sentence?

In standard written Turkish, adverbs like yine (again) usually go before the verb or at the beginning of the sentence. You might hear Hata yaptım yine in casual spoken Turkish, but keeping it at the beginning is the safest and most standard word order.