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Weak almost contact metric manifolds (i.e., the complex struc- ture is replaced by a nonsingular skew-symmetric tensor), defined by the author and R. Wolak, allow a new look at the classical theory and find novel applications in mathematics and physics. An important case of these manifolds, which is locally a twisted product, is a weakβ-Kenmotsu man- ifold defined by the author and D.S. Patra. In the paper, the concept of the ∗-Ricci tensor is adapted to weak almost contact manifolds, the in- teraction of the ∗-η-Ricci soliton with the weak β-Kenmotsu structure is studied and new characteristics of Einstein metrics are obtained. Mathematics Subject Classification. Primary 53C12; Secondary 53C15, 53C25, 53C21.