Weyl sums in Fq[x] with digital restrictions

Let Fq be a finite field and consider the polynomial ring Fq[X]. Let Q ∈ Fq[X]. A function f: Fq[X] → G, where G is a group, is called strongly Q-additive, if f(AQ + B) = f(A) + f(B) holds for all polynomials A, B ∈ Fq[X] with deg B < deg Q. We estimate Weyl Sums in Fq[X] restricted by Q-additive functions. In particular, for a certain character E we study sums of the for