Note on Heath-Brown's estimate for Heilbronn's exponential sum

We show that $S_h(a)=\sum^p_{n=1}e(\frac{an^{hp}}{p^2})\ll (h,p-1)p^{11/12}$, which generalizes Heath-Brown's estimate for Heilbronn's exponential sum $S_1(a)$. We also give a simple proof of a crucial lemma in Heath-Brown's work.