Defeating the Sludge Like is part of exploring Zora’s Domain and “Clues to the Sky” quest in the main story of The Legend of Zelda: Tears of the Kingdom.
This is all part of the lengthy “Sidon of the Zora” questline, which will eventually take you all the way to the Water Temple.
Beating the Sludge Like mid-boss
Once you have discovered the ‘watery bridge’ and found the Land of the Sky Fish and solved the puzzle, return to Sidon at Mipha’s Court to start a battle against the Sludge Like.
This boss is similar to other enemies you may have seen around Hyrule, and it functions the same way, but with a muddy twist. Luckily, Sidon will be here to help out. Talk to Sidon throughout the battle to imbue your attacks with water, and to fire a water projectile ahead. Hit the Sludge Like with the water to rinse it off, allowing you to attack it.
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If you’re farther away, it’ll spit projectiles at you. If it’s close by, it’ll try to chomp on you, stealing your equipped weapon and dealing some damage.
You’ll want to get up close to it, and back away from it when it moves in to bite you. When it misses, its uvula-tongue-dongle will stick out. Hit it with an arrow, polearm, or other weapon that can reach it, to stun it. Once it’s stunned, unleash a flurry of attacks until it recovers. Repeat these steps until it’s dead.
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After it dies, a short cutscene will play and you can claim a (decayed) Royal Claymore from the chest that popped out of the beast’s body.
Sidon will run off to head to the light in the lake, and you should follow him up the stairs of Mipha’s Court. You can leap off the top of the stairs directly to the lake, and glide down to meet up with Sidon. He’ll open up an entryway to the next area, the Ancient Zora Waterworks, which you can hop in whenever you’re ready.
While the next area will provide you with the tools to complete the puzzles, they’re made easier with bomb flowers, if you have any. Feel free to collect some before you enter, if you want an easier time.