Hastur is one of the Great Old Ones and is an Outer God. He can be summoned to earth by someone who has the knowledge of how to hold him back during this time. He can possess a human if they get close enough for too long or he will try and lure them away so that he can take over their body. And no, there are not any specific powers listed because every story involving him varies depending on what was happening in that chapter in the author’s life cycle when it was made.
He governs both dreams and intermittent periods of catalepsy, such as sleepwalking or night terrors for example when a person wakes up terrified after what appears to have just been a deep sleep, but then finds themselves actually still in a deep sleep. He has the ability to cause insanity, as well as suicide if he can get close enough and stay there long enough.