Sums of Distinct Polynomial Residues

Let $p\geq 5$ be a prime.  In 1801, Gauss proved that the sum of distinct quadratic residues modulo $p$ is congruent to $0$ modulo $p$.  A study by Stetson in 1904 showed that the sum of distinct triangular residues modulo $p$ is congruent to $-1/16$ modulo $p$.  Both of these results were extended in 2017 by Gross, Harrington, and Minott, who studied the sum of distinct quadratic polynomial residues modulo $p$.  In this article, we determine the sum of distinct cubic polynomial residues modulo $p$ and prove a conjecture of Gross, Harrington, and Minott.  We further consider the sum of distinct residues modulo $p$ for polynomials of higher degree.