As suggested by the OP, I am making my comment an answer, as it resolves his query in the affirmative.

Theorem 2.1 (Serre, 1991), cited in *Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields* by Peter Beelen, Mrinmoy Datta, Sudhir R. Ghorpadel, gives an explicit formula for the maximal number of solutions in projective $(n-1)$-space of a homogeneous polynomial of arbitrary degree $d$ over finite fields of order $q$.

In particular, this maximum (which perhaps should be called the Tsfasman-Serre-Sørensen number) is: $$ dq^{n−2} + q^{n-3}+q^{n-4}+\cdots+q+1,$$
whenever $d \leq q.$

In the question, $d=p=q$ and so we have a maximum of $p^{n−1} + p^{n-3}+p^{n-4}+\cdots+p+1$ solutions in projective space. But the question was about solutions in affine space. So we scale by $\mathbb{F}_p^*$ then add 1 for the trivial solution to obtain:

$(p^{n−1} + p^{n-3}+p^{n-4}+\cdots+p+1)(p-1)+1=$ $$p^n-p^{n-1}+p^{n-2}=(1−p^{−1}+p^{−2})p^n,$$ as required.