Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of integers in $[1,B]$ which can be represented by $F$ is $\Theta(B^{2/d})$ as $B \rightarrow \infty$, provided that $F(1,0) > 0$.

It is not too much of a leap to guess that we can replace $\Theta(B^{2/d})$ with an asymptotic formula, i.e. there should exist a positive constant $C_F$ such that $$\displaystyle \#\{t \in \mathbb{N} \cap [1,B]: \exists (x,y) \in \mathbb{Z}^2 \text{ s.t. } F(x,y) = t \} \sim C_F B^{2/d}.$$

This fact has been confirmed by C. Hooley for the case $d = 3$ and $$F(x,y) = ax^3 + 3bx^2 y + 3cxy^2 + dy^3$$ irreducible in two separate papers, the first in 1967 and the second in 2000. In the latter, he obtains the constant

$$\displaystyle C_F = \left(1 - \frac{3}{2 \Delta'}\right) \frac{\sqrt[3]{3}\Gamma(1/3)^2}{2 \sqrt[3]{\Delta} \Gamma(2/3)}, $$

where $\Delta$ is an integer such that $D(F) = 81 \Delta^2$, where $D(F)$ is the discriminant of $F$, and $\Delta' = \Delta/G$, where $G = \gcd(b^2 - ac, bc - ad, c^2 - bd)$.

Hooley's work seems to indicate that $C_F$ should exist in general, but probably not very easy to compute. I have not seen any later works where $C_F$ (if it exists) is worked out explicitly.

Has there been further work on investigating what $C_F$ is in other cases, in particular cases with $d \geq 4$? If not, are there any heuristics as to what it should be in general?