Ample vector bundles with small $g-q$

Let X be a smooth complex projective variety and let Z be a smooth submanifold of X of dimension ge 2, which is the zero locus of a section of an ample vector bundle E of rank dim X - dim Z ge 2. Let H be an ample line bundle on X, whose restriction H_Z to Z is generated by global sections. The structure of triplets (X,E,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle E oplus H^{oplus (dim Z -1)} is le q+2, where q is the irregularity of X.