Season short ribs on all sides with garlic powder, kosher salt, and pepper.
Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function according to manufacturer's instructions. Set temperature to 375 degrees F (191 degrees C) if your unit has a temperature setting.
Heat olive oil in the pot. Add short ribs; sear all over until golden brown, about 2 minutes per side. Remove from pot and set aside.
Combine beef stock, tomato paste, carrots, onion, vinegar, and mushroom in the pot. Stir until tomato paste is dissolved. Return short ribs to the pot. Close and lock the lid. Select Meat function according to manufacturer's instructions; set timer for 30 to 40 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, about 15 minutes. Unlock and remove the lid. Remove short ribs and skim any fat from the top of the liquid in the pot.
Stir water and cornstarch together to make a slurry. Set pressure cooker to Saute and bring the liquid to a boil. Stir in the slurry until mixture is thickened to your liking. Season with salt and pepper and return the short ribs. Cook on low until flavors come together, about 5 minutes more.
