In this paper, we generalize the block positive-semidefinite splitting preconditioner (Cao, J. Comput. Appl. Math., 374, 112787, 2020) by introducing a new parameter for generalized saddle point linear systems and the spectral distribution of the corresponding preconditioned matrix is analyzed. We show that the corresponding preconditioned matrix has clustered eigenvalue distribution when the iteration parameters tend to be 0. We introduce some numerical experiments to demonstrate the validity of the presented theoretical results.