Generalized polarized manifolds with low second class

On a smooth complex projective variety X of dimension n, consider an ample vector bundle E of rank ran geq 2 and an ample line bundle H. A numerical character m_2 =m_2(X,E,H) of the triplet (X,E,H) is defined, extending the well-known second class of a polarized manifold (X,H), when either n = 2 or H is very ample. Under some additional assumptions on F : = E oplus H^{oplus (n-r-2)}, triplets (X,E,H) as above whose m_2 is small with respect to the invariants d := c_{n-2}(F) H^2 and g := 1 + (1/2)(K_X+c_1(F)+H) cdot c_{n-2}(F) cdot H are studied and classified.