A note on the fourth power mean of the generalized Kloosterman sums

In the paper [1], the first author used the analytic methods and the properties of Gauss sums to study the computational problem of the fourth power mean of the generalized Kloosterman sums for any primitive character χmodq, and give an exact computational formula for it. In this paper, we considered the same problem for non-primitive character χmodq, and solved it completely.