This study is concerned with the problem of finite-time filter design for a class of Itô stochastic systems with Markovian switching and distributed time-varying delays. Firstly, a partially mode-dependent filter is designed to accommodate to unreliable network transmission. The attention is focused on deriving sufficient conditions for the filtering error system to ensure the finite-time boundedness and to satisfy a prescribed disturbance attenuation. Then based on stochastic functional theory, the existence of filter is presented by solving existing linear matrix inequalities optimisation problems. Furthermore, the result is extended to the case where the mode information is completely transmitted. Finally, a numerical example is provided to show the effectiveness of the proposed results.