Image of $(\Bbb Z/p\Bbb Z)^\times$ under a polynomial map.

Question

Consider the following result:

Proposition. For an odd prime $p$ such that $\left(\frac{-1}{p}\right)=-1$, every square in $(\Bbb Z/p\Bbb Z)^\times$ is of the form $x^3+x$.

I am trying to give a general formulation for such phenomenon, namely

1. Given an odd prime $p$, characterize the polynomials $f\in\Bbb Z[x]$ such that the image of $h_f: (\Bbb Z/p\Bbb Z)^\times\to (\Bbb Z/p\Bbb Z)^\times, h_f(x)=f(x)$ contains the group of quadratic residues mod $p$.

2. Given a polynomial $f\in\Bbb Z[x]$, characterize the odd primes $p$ such that the image of $h_f: (\Bbb Z/p\Bbb Z)^\times\to (\Bbb Z/p\Bbb Z)^\times, h_f(x)=f(x)$ contains the group of quadratic residues mod $p$.

Note: The proposition on the top is equivalent to the following statement related to the number of solutions a polynomial congruence:

There are exactly $p-1$ solutions to the equation $x^3+x=y^2$ in $(\Bbb Z/p\Bbb Z)^\times$ for odd prime $p = 3 \bmod 4$.