In this article, the problem of constant yet deferred output constraint for uncertain strict-feedback nonlinear systems is studied. By ``deferred output constraint,'' we mean that the system output is free/released from any constraint in the initial interval and then preserves within a bounded region right after a finite time. Due to such a form of output constraint, the normally employed Barrier Lyapunov Function (BLF)-based results become invalid because the corresponding BLF is undefined in the initial period. The problem will be rather complicated yet challenging if computation and communication constraints are taken into account. By developing an error-based nonlinear function and constructing a prescribed-time scaling function, together with the approximate ability of neural networks, a varying threshold-based event-triggering adaptive neural control algorithm is presented such that not only the deferred output constraint can be ensured and the network resources can be saved but also the tracking error is able to converge to a pregiven region in a prescribed time. Simulations are provided to demonstrate the effectiveness of the proposed control.