The problem of almost disturbance decoupling with internal stability (ADD) is formulated, in terms of an L-2-L-2p (instead of an L-2) gain, for a class of high-order nonlinear systems which consist of a chain of power integrators perturbed by a lower-triangular vector field. A significant feature of the systems considered in the paper is that they are neither feedback linearizable nor affine in the control input, which have been two basic assumptions made in all the existing ADD nonlinear control schemes. Using the so-called adding a power integrator technique developed recently in [15], we solve the ADD problem via static smooth state feedback, under a set of growth conditions that can be viewed as a high-order version of the feedback linearization conditions. We also show how to explicitly construct a smooth state feedback controller that attenuates the disturbance＇s effect on the output to an arbitrary degree of accuracy, with internal stability.