Hua andChen gave estimates of sums\(\sum\limits_{x = 1}^q {e(\mathfrak{F}(x))} \) wheree(z)=e2πiz and\(\mathfrak{F}\) is a polynomial of the typef(x)/q wheref(x)=akxk+...+a1x with integer coefficients having gcd (q, ak,...,a1)=1 But no good estimates hold for these sums whenq is small in comparison tok. We therefore consider here a related but different class of polynomials. Special emphasis is given to the cubic case.