Under consideration is the discrete three-field Blaszak-Marciniak lattice equation. Firstly, this discrete equation is mapped to the continuous nonlinear equations under the continuous limit. Secondly, the generalized $(m,3N-m)$-fold Darboux transformation of this discrete equation is constructed and established. Finally, by applying the resulting Darboux transformation, some singular rational solutions and mixed exponential-rational solutions are presented, in particular, their limit state analysis and singular trajectories are analyzed graphically. These results may be helpful to explain some relevant physical phenomena.