This work addresses the distributed consensus tracking problem for an extended class of high-order nonlinear multiagent networks with guaranteed performances over a directed graph. The adding one power integrator methodology is skillfully incorporated into the distributed protocol so as to tackle high powers in a distributed fashion. The distinguishing feature of the proposed design, besides guaranteeing closed-loop stability, is that some transient-state and steady-state metrics (e.g., maximum overshoot and convergence rate) can be preselected a priori by devising a novel performance function. More precisely, as opposed to conventional prescribed performance functions, a new asymmetry local tracking error-transformed variable is designed to circumvent the singularity problem and alleviate the computational burden caused by the conventional transformation function and its inverse function, and to solve the nondifferentiability issue that exists in most existing designs. Furthermore, the consensus tracking error is shown to converge to a residual set, whose size can be adjusted as small as desired through selecting proper parameters, while ensuring closed-loop stability and preassigned performances. One numerical and one practical example have been conducted to highlight the superiority of the proposed strategy.