Go to the Courts tab in Manage events. This screen is mostly about how you can curate the court bookings your club can manage for the event. But, more generally, it covers how your club interacts with the event and one of these aspects is whether you wish the event to show on your kiosk screen as a quick link.
To choose this event to show as a quick link on the kiosk find the panel that is titled Kiosk screen configuration and check the Show quick link on kiosk screen box.
Note that only events that have draws (leagues, tournaments etc) may be chosen to have a quick link.
You'll probably want to check this item for your monthly leagues. But it may also be good to allow members to quickly access a club championship draw in this way as well.