There are 25 hidden competitive minigames in It Takes Two, two of them in the Cuckoo Clock. They will all be marked with a hovering tambourine, and finding all 25 will get you the Minigame Megalomania achievement or trophy. You can then access all of these minigames at any time from the menu — the following is in order of how the games are listed in the menu, slightly different from the order in which you find the games. Both minigames in Cuckoo Clock are in the town that you explore in the extended Gates of Time segment.
In the far left area of the town, look for a giant lever; the red neon Helltower sign is a good point of reference. Pull the lever together, and both May and Cody will have to parkour their way through the town with rope swinging and rail grinding to hit orange ring checkpoints and extend their timer — whosever timer hits zero first will explode.
In the far back section of the town, look for a display that depicts a horse derby. Cody and May will control the horses — hit the jump button or down on the control stick to duck whenever low or high obstacles are in your way. Failing to do so will slow you down and potentially cost you the race.