How you can express "ugh" in Mandarin?
I mean, when you're trying to express disgust or annoyance, what's the proper interjection to use?

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You can just use "ugh"(呃 in Chinese).
This interjection is also widely used in Chinese.

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哇 = wa

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呵呵 sarcastic laugh, used to.express contempt

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It depends on different people. but mostly used：啊！to express surprise or something you cannot stand.     呸！is a little bit bad language to express disdain.   哎呀：most girls like to use it to express annoyance, boys do not use them ofen, because it sounds like a girl.  哎/唉 to express disappointed or tired.   You can just use the words you're used to. I think people understand them by tone, not the word itself.

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呸/哎/啊