The sentence could be rephrased as follows:
How important is it for you to have a part of your week for your
...to have time for your free-time activities?
I'm not a native speaker of English, too, so I may be wrong :) I have found an answer to the question on the internet:
Examiner: How important a part of your week are your free-time
Candidate: Actually, not really that important. Perhaps if I was
working, my time off would be the most important part of the week in
some ways, but because I'm a student, I don't so much feel the need to
let my hair down at the weekends. I enjoy most of the time I'm
studying, too. It's not like doing a stressful job.
The candidate's answer words "not really that", "the most" suggest that the question is about the level of importance.