In this paper, we study structures of smooth complex projective polarized manifolds (X, H) of dimension n ≥ 2 which are rationally connected with respect to a family of H-degree four. Under the assumption , we prove that, with two kinds of exceptions, the Picard number of X is at most four and X is covered by rational curves of H-degree one. In addition, we provide a classification in case n = 2.