We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $K$-contact structures and allow a new look at the classical theory. We characterize weak $K$-contact manifolds among all weak contact metric manifolds using the property well known for $K$-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $K$-contact manifold. We also find sufficient conditions for a weak $K$-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.