Quick answer: “Good luck!” (Or, “Break a leg!” in theater.)
Longer explanation:
There isn’t a good literal translation for most cases, because the actual words do mean “hang in here”, “do your best”, “work hard”, etc.
However, as Robert Whiting notes in the definitive book on gaijin players in Japanese baseball, You Gotta Have Wa, it’s often said to professional athletes, and when the interpreters would translate that into something like “Do your best!” the Western players would universally react negatively because it’s insulting to tell a professional to best because it’s assumed that professionals will do their best and all that remains is wishing them good luck.
After the interpreters discovered this cultural difference and started saying “Good luck!” the gaijin player’s face light up, they would smile and say thank you, similar to what a Japanese player would do when told to “gambaru.”
It’s a cultural transaction, where the speaker is conveying a cultural expectation to the listener, rather than providing lim
As I explain to people, this can be generalized to adults, because it’s the grown up thing to do one’s best and all that people need is a bit of luck.
Coaches (and overbearing sports parents) can encourage their athletes (children) to excel, but one of the few times this would be appropriate.
I participate on a Japanese / English language exchange site and Japanese will often say “I’ll work hard at my job today.” 今日、仕事を頑張ります。Kyo, shigoto wo gambarimasu . but that’s not something a native English speaker would say. There may be particular times when we would say, “I have to bust @ss today” but that’s an exception rather than a rule.
Polite fictions: why Japanese and Americans seem rude to each other is an older book, but it also mentions this phrase and covers the cultural aspects of it. (IIRC)
Another one is otsukaresama, “honorable tired person” and that’s not transalatable, either.