No weekly competition
There's no organized weekly competition on Ktkarena: no theme of the week, no dedicated leaderboard, no badge or bonus for the best.
The weekly missions
In the app, open Profile → Missions, “Weekly” tab. Each mission you complete earns you XP, which you collect by tapping “Collect”. The screen shows when the missions reset.
The “Week” view of the leaderboard
The leaderboard (Profile → Leaderboard) has a “Week” view, next to “Day”, “Month” and “All”. It doesn't come with a prize.