Mhmm. It should be "Do you have time during the week next week?" or "Next week, do you have any time on a weekday?"
It's fine for me. That's exactly what I entered. Although, my answer wasn't accepted for some reason.
"Do you have time next week on the weekdays?" is apparently not correct. But 평일 means "weekday". LOL!
The default English translation is not intuitive or easy to parse. "Do you have time on a weekday next week?" resolves the ambiguity that comes from having semantically divergent uses of "week" (M-F vs. S-S) in neighboring time phrases.