In his 1952 AMS Bulletin article "The Riemann zeta and allied functions" Chowla asks the following:

Given a polynomial $f$ with integer coefficients, how many residue classes mod $p$ does its image occupy?

Chowla notes without references that Davenport has proven that for degree $4$ polynomials the number of residue classes is asymptotic to $5p/8$. My question is what is currently know about this problem?

Certainly this is reminiscent of Weil's theorem about the square root cancellation of polynomial exponential sums, but I don't see how to deduce a solution from Weil's estimate.