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A weak metric $f$-structure $(f,Q,\xi_i,\eta^i,g)\ (i=1,\ldots,s)$,generalizes the metric $f$-structure on a smooth manifold, i.e., the complexstructure on the contact distribution is replaced with a nonsingularskew-symmetric tensor $Q$. We study geometry of a weak $f$-K-contact structure,which is a weak $f$-contact structure, whose characteristic vector fields areKilling. We show that $\ker f$ of a weak $f$-contact manifold defines a$\mathfrak{g}$-foliation with an abelian Lie algebra. Then we characterize weak$f$-K-contact manifolds among all weak metric $f$-manifolds by the propertyknown for $f$-K-contact manifolds, and find when a Riemannian manifold endowedwith a set of orthonormal Killing vector fields is a weak $f$-K-contactmanifold. We show that for $s>1$, an Einstein weak $f$-K-contact manifold isRicci flat, then find sufficient conditions for a weak $f$-K-contact manifoldwith parallel Ricci tensor or with a generalized gradient Ricci solitonstructure to be Ricci flat or a quasi Einstein manifold. We prove positivedefiniteness of the Jacobi operators in the characteristic directions and usethis to deform a weak $f$-K-contact structure to an $f$-K-contact structure.Finally, we define an $\eta$-Ricci soliton structure on a weak metric$f$-manifold and find sufficient conditions for a compact weak $f$-K-contactmanifold with such a structure of constant scalar curvature to be$\eta$-Einstein.

Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds