The existence of solutions of a Diophantine exponential equation

Question

Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation $$p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$$ with the restrictions $x_i<n$ for all $i$.

Observe that for the numbers $x_0=\dots=x_{p^n}=n$ and $x=2n$ we have $$\sum_{i=0}^{p^n}p^{x_i}=(p^n+1)\cdot p^n=p^{x}+p^n$$so, the restriction $x_i<n$ is imposed for ruling out this obvious solution.

Question 1. Determine $p$ and $n$ for which the Diophantine exponential equation $p^x+p^n=\sum_{i=0}^{p^n}p^{x_i}$ does not have positive integer solutions $x,x_0,\dots,x_{p^n}$ with $x_i<n$?

Question 1 can be also reformulated as follows.

Question 2. Determine $p$ and $n$ for which there exists a polynomial $f(x)$ of degree $<n$ with non-negative integer coefficients such that $f(0)=0$, $f(1)=p^n+1$ and $f(p)=p^n+p^x$ for some $x>n$.

Question 3. What is the answer to Questions 1,2 for $p=2$?

Answer 1

This is possible for all $n\geq 2$. We have $$p^{n+1} + p^n = p (p^n+1) + (p^2-p) (p^{n-2}+ p^{n-3} + \dots + 1) $$ so for question 2 the polynomial $$f(x) = (p^n+1 )x +(p^{n-2}+ p^{n-3} + \dots + 1) ( x^2-x)$$ works. This polynomial certainly has nonnegative coefficients since $p^{n-2}+ p^{n-3} + \dots + 1 < p^{n-1} < p^n+1$.

For $n=1$ it is of course impossible since there are no positive integers $x_i<n$.