We will show that every group admitting an unbounded quasi-cocycle into a mixing non-amenable representation is not inner amenable. We will discuss how our approach generalizes a recent result of Dahmani, Guirardel, and Osin on the non-inner amenability of so-called acylindrically hyperbolic groups. Other applications such as to central sequences in group-measure space algebras as well as some open problems will also be touched upon. This is based on joint work with Ionut Chifan and Bogdan Udrea.