여쭈다 is the humble word for asking, used when the person you are asking outranks you; the plain word is 묻다. -는지 wraps the question up so it can sit inside a polite frame, and most of these lines land on 여쭤봐도 될까요 or 여쭤봐도 돼요, 'may I ask...'. Most also open with 혹시, softening it further to 'if you don't mind my asking'. This is interview and first-meeting language.