Weyl sums in Fq[x]Fq[x] with digital restrictions

Let FqFq be a finite field and consider the polynomial ring Fq[X]Fq[X]. Let Q∈Fq[X]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)f(AQ+B)=f(A)+f(B) holds for all polynomials A,B∈Fq[X]A,B∈Fq[X] with degB