The introduction of a kernel into quasilinear Hardy-type operators is an important direction. However, obtaining results for the general type of kernel is an undefined problem, so researchers study Hardy-type operators for certain kernel classes. Generalizing the research, we obtained the criteria for the validity of quasilinear integral Hardy-like inequalities belonging to the classes  and  in the parameter ranges . That is, necessary and sufficient conditions for the validity of quasilinear operators involving the kernel Hardy operator in weighted Lebesgue spaces were determined. This allows us to study generalized bilinear operators. In particular, the results of quasilinear inequalities with kernels provide alternative method for estimating bilinear inequalities. This study contributes to the theory of integral operators in functional analysis and can be used in finding solutions to differential equations.