Learn Danish: Sektoren mangler folk. - The sector is short of people.

No, they are false friends. In Danish, the verb "at mangle" means to lack, to need, or to be missing. Instead of using a descriptive phrase like "is short of", Danish phrases this actively, so the sentence literally translates to "The sector lacks people".

Why doesn't the word "folk" have a plural ending in this sentence?

When used to mean "people" in a general, uncountable sense, "folk" is an irregular neuter noun that takes no ending in the indefinite plural. It looks exactly the same as its singular form "et folk", which translates to a nation or a distinct population.

Could I use the word "mennesker" instead of "folk" to say "people" here?

You could, but "folk" is much more natural in this context. The word "mennesker" tends to emphasize humans as individuals or biological beings, whereas "folk" is the standard choice when talking about workers, crowds, or people in an everyday societal context.

How flexible is the verb "mangler"? Could I rephrase this so "folk" is the focus?

Yes, "at mangle" is incredibly flexible. If you want to shift the focus, you can use the existential "der" and say "Der mangler folk i sektoren" (There is a lack of people in the sector). Both that structure and the active phrasing in our sentence are extremely common at the B2 level.