Acoustic black hole (ABH) structures have been developed as a new approach for manipulating flexural wave propagation in plate and beam structures. To overcome the limitation of single ABH cell on manipulating wave propagation as the results of the coupling between the size of ABH cell and wave length, this work investigates a thin plate with periodic array of two-dimensional (2D) imperfect ABH structures for wave manipulation. Dispersion relations and eigenstates of an infinite periodic lattice structure, which is consisted of a square array of imperfect ABH indentations, are numerically investigated. Both of them are shown to be different from those of the infinite periodic lattice structure based on traditional ABH structure. The results show that the considered periodic lattice structure based on a generally power-law-profiled indentation can generate remarkable dispersion relations. Additional insight on wave manipulation properties of these periodic lattice structures is analysed by using steady state distributions of wave field in thin plates coupled with these periodic lattice structures. The distributions of wave field demonstrate some special field patterns: wave front reshaping and directional wave propagation.