A new system of the set-valued mixed quasi-variational-like inclusions (SSMQVLI) involving H_η-monotone operators is studied in general Banach spaces without uniform smoothness. By using the resolvent operator technique of H_η-monotone operators, a new iterative algorithm for finding approximate solutions to SSMQVLI is proposed. It is shown that the iterative sequences generated by the algorithm converge strongly to the exact solution of SSMQVLI under appropriate assumptions. These obtained new results have extended and improved previous results.