Walkthrough: Once you reach the island habitat with both Mario and Cappy, you will be tasked with searching for and obtaining five Moon Shards that are scattered throughout the island, the same that will then be used to form another Power Moon. The first Moon Shard can be found by the nearby Moe-Eyes, creatures that are practically harmless to you. The second Moon Shard can be found by going to the left of the first shard, all the way up to the top of the hill on the edge of island near another stone pillar.
The third Moon Shard can be found in the opposite corner of the island, the south eastern edge between two cacti. After obtaining the first three Moon Shards, return to the toxic pool in the middle of the island. Use Cappy to capture one of the Moe-Eyes, and then use his sunglasses to spot an invisible bridge across the toxic leading to the next Moon Shard. There will also be a bridge nearby that leads to more Purple Coins.
After obtaining the fourth Moon Shard, head back to the northwest corner of the island to find a moving platform that will bring you up yet another stone pillar. As you ride it, capture another Moe-Eye with the help of Cappy, now seeing invisible platforms that lead to the next pillar that holds the final Moon Shard. After collecting all five Moon Shards, a Power Moon will appear at the edge of a destroyed bridge on the northern part of the island. Grabbing it will power up more moving platforms that lead you off the island.