The weak nonlinear regime of the high-frequency modulated pulse propagation is considered in a planar graded-index structure, with its longitudinal inhomogeneity and the presence of the cladding being taken into account. The model equation is formed by completing the standard nonlinear wave equation with a term that adequately characterizes the material dispersion of dielectric material filling the waveguide. An asymptotic solution to the model equation is constructed with respect to a small parameter prescribing the order of magnitude of the wave field, and the second power of this parameter - the ratio of scales of longitudinal and transverse inhomogeneities. Explicit analytical formulae are obtained for the propagation constants and the transverse field distributions of various modes for the hyperbolic secant profile. For the pulse envelope the nonlinear Schrödinger equation is derived, its coefficients depend on the longitudinal coordinate, differ for different modes, and account for material and waveguide dispersions. Effects caused by the presence of the cladding and material dispersion are estimated separately.