Location: The Dalimil Inn
Petitioner: Xaver, Bathhouse Owner
Unlocked: Complete Follow the Crystals
Reward: The Breath of Fire (Ignition), 1 Scarletite, 25 Renown
In order to unlock this side quest you must first begin the main story quest Riddle of the Sands, which will take you to the Dalimil Inn, in the Velkroy Desert. Partway through Riddle of the Sands you'll have to complete a sub-quest, Follow the Crystals, and doing so will earn you a seal. This will unlock Hot Water.
Given the name of this quest it shouldn't be any surprise that the petitioner is Xaver, the owner of the Baths. Enter the Baths, on the other side of the bridge in the middle of Dalimil, and speak to Xaver. He's freaking out because the water is suddenly way too hot, and he doesn't know why. He begs Clive to check out the source of the water.
Agree and you'll be pointed towards Doeznov Terraces, the winding path to the south of the Dalimil Inn. Follow the path south through the terraces until you hit a three-way intersection. Take a right and you'll find Fangslip, the springs that supply the water for the Baths. Normally this place is inhabited by Helldivers, but right now they're occupied by a large group of Bath Bombs.
Given how volatile Bombs tend to be you want to be careful not to set too many of them off at once. Try to draw the Bombs away from each other and use Eikonic Abilities to quickly wipe them out. If there are any signs of an explosion, steer clear. Wipe out the Bath Bombs and a blue Bomb called Old Faithful will appear, along with two more Bath Bombs. This one uses more fire attacks, and has a lot more HP. Take out the normal Bath Bombs so you can focus on Old Faithful.
Defeat the Bombs and you'll receive Wyrrite, Magicked Ash, and Black Blood. Jump back to the Baths and speak to Xaver for the rest of your reward.