As a native speaker of British English I have always said "at the weekend" and "at weekends". I can understand speakers of other varieties of English who use "on" rather than "at" but it sounds odd to me.
You may not still be active but I wanted to ask this same question...glad you asked and I’m glad luless clarified. I had not “consciously” been listening to the pronunciation of weekend. Interesting rule to keep in mind.
Now you tell me how it is so that "I have time for cooking on weekends only" is counted as a mistake, but the presumably correct answer the application suggests is "I only have time for cooking on weekends".