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I don't know about Korean but the Japanese, with whom the Koreans share similarities, have the phrase "yoroshiku onegaishimasu". This phrase is used when entrusting the other person with some task or responsibility. As such, it can be used in an introduction where it would mean something like: "please treat me well"; though it would usually be translated to a more common greeting like "Nice to meet you".
In cases like the ones in your screenshots it would mean something like: "Please perform the task well", where a more natural translation might be: "I'll leave it to you" or even "good luck" (as an expression of the desire that the task goes well).
If the Korean phrase is any similar, then the translation is wrong due to ignorance about it's contextual sensitivity.