In the paper under review, the authors prove that for any odd prime \(p\), \[ C_4(p):=\sum_{m=0}^{p-1}\left|\sum_{n=0}^{p-1}\mathrm{e}\left(\frac{n^2(m+n)}{p}\right)\right|^4=2p^3+O(p^{5/2}), \] where \(\mathrm{e}(x)=e^{2\pi ix}\). They consider two cases when \(p-1\) is divisible by \(3\) or not. When \(3|(p-1)\), they provide the following explicit and better approximation for \(C_4(p)\), \[ C_4(p)=2p^3-p^2\cdot\left(4-\left(\frac{-3}{p}\right)\sum_{a=1}^{p-1}\left(\frac{a+a'-1}{p}\right)\right), \] where \((\frac{\cdot}{p})\) denotes the Legendre’s symbol modulo \(p\), and \(a'\) is the multiplicative inverse of \(a\) modulo \(p\).