Additive character sums of polynomial quotients

Let p be a prime, f (X) is an element of Z[X] with leading coefficient not divisible by p, and R a complete residue system modulo p. For an integer u we define the polynomial quotient q(f,p,)R(u) f(u) - f(p,R) (u) /p mod p, 0 <= q(f,p,)R (u) < p, where f(p,R)(u) f (u) mod p and f(p,R)(u) is an element of R. Among the polynomial quotients are the well-studied Fermat quotients q(p) (u) u(p-1) - u(p(p-1)) / p mod p. We study additive character sums of s-dimensional vectors of polynomial quotients which are nontrivial if either the degree of f(X) is small or f (X) is a monomial of large degree smaller than p. For s = 1 and u(w) - u((p-1)w) /p mod p with a large gcd(w, p - 1) we obtain much stronger bounds by a reduction to the Burgess bound.