Affine monodromy and exact value distributions over finite fields

We study finite field value distributions through the fixed-point statistics of monodromy groups. For a regular degree-$n$ cover, omitted values are controlled by derangements. Thus natural symmetric monodromy gives support density $1-D_n/n!\to 1-e^{-1}$, while the Cameron--Cohen bound gives the universal ceiling $1-1/n$, attained by sharply $2$-transitive affine monodromy. We give an explicit polynomial realization of this optimal mechanism. For $N=p^e$ and $h\mid N-1$, set $$\Lambda_{N,h}(U)=U\bigl(U^{(N-1)/h}-1\bigr)^h .$$ Its geometric Galois closure is rational, $$U=z^h, \qquad T=(z^N-z)^h,$$ and its geometric monodromy is the affine group $(\F_N,+)\rtimes H_h$. For every extension $\F_Q/\F_p$ we compute the complete fibre enumerator of $\Lambda_{N,h}$ exactly, including the nonregular cases. In the full affine case $h=N-1$, the polynomial $$U(U-1)^{N-1}$$ attains the Wan--Shiue--Chen upper bound for non-permutation polynomials over every finite field containing $\F_N$; over arbitrary extensions we compute the exact defect from that bound.