We consider a weakly nonlinear process of propagation of a modulated pulse with high-frequency filling in a plane graded-index waveguide layer. The transverse distribution of the refractive index is specified by a function of the hyperbolic-secant type, which is nonsymmetric relative to the layer axis. Due to this fact, the analysis includes not only the asymmetry of the waveguide layer, but also different properties of the cladding on both sides of the layer. Longitudinal inhomogeneity of the studied waveguide structure is also admissible. The transcendental characteristic equation is derived to determine the mode propagation constants, and the transverse wave fields are expressed in terms of a hypergeometric function. Numerical calculations demonstrate the possibility of the mode cutoff due to the asymmetry of the waveguide layer. The nonlinear Schrödinger equation is derived for the pulse envelope with the coefficients corresponding to the nonsymmetric waveguide layer.