In this paper, we classify $(k,\mu)'$-almost Kenmotsu manifolds admitting some special vector fields such as concircular and torse-forming. Furthermore, we characterize $(k,\mu)'$-almost Kenmotsu manifolds with an $\eta$-Ricci soliton whose potential vector field is projective and affine conformal. Beside these we study gradient $\eta$-Ricci soliton on $(k,\mu)'$-almost Kenmotsu manifolds. Finally, the existence of an $\eta$-Ricci soliton on a 3-dimensional $(k,\mu)'$-almost Kenmotsu manifold is ensured by a proper example.