In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from ℂm into an algebraic variety V intersecting a finite set of fixed hypersurfaces in subgeneral position. As an application, we prove a result on algebraic degeneracy of holomorphic curves on \({\cal P}_c^1\) intersecting hypersurfaces and difference analogue of Picard’s theorem on holomorphic curves. Furthermore, we obtain a second main theorem of meromorphic mappings intersecting hypersurfaces in N-subgeneral position for Veronese embedding in ℙn(ℂ) and a uniqueness theorem sharing hypersurfaces. Our second main theorem and difference analogue of Picard’s theorem recover the results of Cao-Korhonen [1] and Halburd-Korhonen-Tohge [8], respectively. By a way, we also obtain uniqueness theorems of meromorphic mappings which improve the result of Dulock-Ru [4].