The authors investigate small solutions of diagonal congruences of the form \[ \sum_{i=1}^n a_ix^k \equiv c \bmod{p}, \] where \(p\nmid a_i\) for \(i=1,\dots,n\). This problem has received much attention over the years. Now the authors obtain a best possible result in this direction, improving upon their previous results [Funct. Approx. Comment. Math. 56, No. 1, 39–48 (2017; Zbl 1434.11089)]. What they establish is that if \(n>\frac{3}{2}(k^2+k+2)\), then a solution \((x_1,\dots,x_n)\) with \(1\leq x_i \ll_k p^{1/k}\) for \(i=1,...,n\) exists. A large part of this paper is devoted to proving a sharp bound for the number of solutions to the above congruence in general cubes. The authors link this problem to integer solutions of systems of the form \[ \begin{aligned} x_1+ \cdots +x_n&= y_1+ \cdots +y_n\\ x_1^2+ \cdots + x_n^2&= y_1^2 + \cdots +y_n^2\\ \dots\\ x_1^k+ \cdots + x_n^k&= y_1^k + \cdots +y_n^k, \end{aligned} \] which come up in connection with Vinogradov’s mean value theorem. They use the recent results by J. Bourgain et al. on the number of integer solutions of systems of this form [Ann. of Math. (2) 184, No. 2, 633–682 (2016; Zbl 1408.11083)].