Translation:Her husband will be in New York next week.
when we're talking about the future, I think 会 always translates to "will." However, 会 can also be a verb that means "to know (how to)" which is maybe where you're confusing about "can" comes from? For example 我会跳舞 means "I know how to dance/I can dance" but it's a different meaning altogether.
I think it also depends on the context. Swansae's example 我会跳舞 certainly means "I know how to dance" by itself, but i think it could also mean "I will dance" if it is, say, the answer to the question "What will you do tomorrow at the talent competition?"
That said, when combined with the word 在, which is like the prepositions in/at/on, 会 usually means "will" rather than "to know how to," as it doesn't make much sense to say "I know how to be in New York." To express "I can be in New York," I think one would use 可以 instead.
Pretty sure 'hui' never means 'can' when paired with positional/travel/existential type verbs that aren't skills/actions. Will go, will be, will arrive vs will/can walk, will/can fly, will/can run. Perhaps it's assumed one always knows how to exist and be places. "Know(s) how" is a probably a more precise translation than just "can." 'Can go' would be 'ke yi qu.'
I answered "will be able to be in New York" because I have know 会's meaning is "can". The "correct answer here was "I am going to be in NY'. I don't know if I'd get it correct if I put "will be in NY", but the reason why "I'm going to be" is chosen as the correct one is because the speaker has a firm-sounding plan (= I know I'll be there) to do that, which sounds a bit firmer that saying "I'll be there", I assume. Is that correct?