This work presents a contribution to the development of computational homogenization procedures, in which a Cosserat continuum at the macroscopic level and an elastic heterogeneous Cauchy medium with periodic texture at the microscopic level are coupled. A new polynomial kinematic map is formulated, taking into account all the Cosserat deformation components and satisfying all the governing equations at the micro-level in case of a homogeneous elastic material. In addition, the distribution of the perturbation field, arising when the actual heterogeneous nature of the material is taken into account, is analyzed. Contrary to the case of first-order homogenization, it emerges that the periodicity conditions are no longer appropriate. A micromechanical method is finally addressed to solve this microstructural boundary value problem.