Very useful phrase, thank you for putting it in! ☺
Just a comment on the suggested translation, 'I have time to cook only on weekends'. I can't imagine a native English speaker ever saying it in this way; to my mind it feels a bit like 'I actually do have the time, but I've more important things to do in the week than cooking, so I only cook on weekends' rather than 'I really don't have the time to cook, except for on weekends'.
I wrote: 'I only have time to cook on weekends', which was also accepted. This order seems much more natural to me. Sorry if this seems like hiperpoprawność, the suggested sentence in English just really bugs me
I believe the Polish word should rather be "pedantyzm", "hiperpoprawność" is "hypercorrectness" and I'd associate it mostly with pronunciation ;)
About the English word order... okay, my team agrees with you, but the problem is that if I put your version as the main one, then most learners would try to answer "Mam tylko czas..." or "Tylko mam czas..." and none of those sound like natural word orders to me. I think it's one of those cases where if the English sentence isn't exactly the most natural - but isn't wrong - I'd rather sacrifice the naturalness of English for the sake of teaching Polish.
I'm also simply surprised that "only on weekends" sounds strange to natives and in fact the interpretation that you give really sounds natural to me, it's not likely to me that I literally have no time, I just probably have better things to do.