The Soul Essence Gem is an item that is used to hold part of the Dragonborn's soul if the option to become partially soul-trapped is chosen at the end of Chasing Echoes.
When found, it will remove the Weakened Soul ability that is gained upon becoming soul trapped.
Found in the Soul Cairn, in a large castle-like structure. Once picked up, it changes into a black soul gem.
After retrieving the Elder Scroll, Valerica can be asked to give the Dragonborn a miscellaneous quest (including quest marker) to find the Soul Essence Gem.