Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function. Add oil; when hot, saute sausages and onion together until browned, 5 to 7 minutes. Transfer to a bowl.
Mix chicken broth, turnip greens, great northern beans, cannellini beans, tomatoes, water, ham bone broth, garlic soup mix, Cajun seasoning, and garlic powder together in a large bowl.
Pour broth mixture into the hot pressure cooker and scrape the browned bits off the bottom. Return cooked sausages and onion to the pressure cooker. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 15 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove lid.