This paper is concerned with the issues of stability analysis and control synthesis for a class of linear stochastic switching systems in discrete-time domain. The switching dynamics are considered to be governed by a semi-Markov process and the sojourn time for each system mode is deemed to be finite. A mode-dependent control scheme is employed with an adaptation sense in the presence of time delays in the mode switching of controller, which is manifested as a constant lag between the system mode and the controller mode. A novel form of Lyapunov function is adopted, in which the Lyapunov matrix depends on the modes of both the system and the controller as well as the time since the occurrence of the last mode switching. On the basis of the new proposed sigma-error mean-square stability that integrates the weights of all the system modes, numerically testable stability criteria are developed via the semi-Markov kernel approach. In virtue of certain techniques that can eliminate the terms containing powers or products of matrices, a desired mode-dependent stabilizing controller is designed such that the closed-loop system is sigma-error mean-square stable by allowing a mode-unmatched controller to perform before the controller switches to a mode-matched one. Finally, the theoretical results are applied to a practical example of one joint of a space robot manipulator to demonstrate the effectiveness, applicability and superiority of the proposed control strategy as well as the necessity of considering the mode-switching delays in the designed controller. (C) 2018 Published by Elsevier Ltd.