  The first-order correction to the one-form sector of equations of the $3d$
higher-spin theory is derived from the generating nonlinear HS system by virtue
of the shifted homotopy approach. The family of solutions to the generating
system that disentangles equations for dynamical and topological fields in the
first order of perturbation theory is found. This family is shown to belong to
the different cohomology class compared to the solution found earlier by the
direct methods. The related cohomology is shown to be the same as that
underlying the mass deformation in the matter sector of $3d$ higher-spin
equations.

High Energy Physics - Theory