In this paper, the problem of controller design for a class of switched nonlinear time-delay systems with finite-time output constraints is studied. In order to limit the output tracking error to a predetermined boundary in finite time, an improved performance function, finite time performance function(FTPF), is introduced. In the process of controller design, the barrier Lyapunov function is used to solve the problem of output constraint. Then, a new adaptive multi-dimensional Taylor network(MTN) control method is proposed by combining the universal approximation property of the MTN with adaptive backstepping technology. In this method, Lyapunov-Krasovskii functionals are designed to ensure the stability of the switched system in the presence of time-delay. Finally, two simulation examples are given to illustrate the effectiveness and practicability of the design scheme.