For each prime $p$, let us define $$w_p:=|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|,$$ where $a\pmod p$ denotes the residue class $a+p\mathbb Z$.
Based on my computation, I conjecture that $$w_p=(1-e^{-1})p+O(\sqrt{p}).\tag{1}$$ For the $2\times 10^6$-th prime $p=32452843$, we have $$w_p=20519206,\ \ w_p-(1-e^{-1})p\approx 5096.75\ \ \text{and}\ \ \sqrt{p}\approx 5696.74.$$
Question. Does $(1)$ hold? If it is true, how to prove it?
Your comments are welcome!