FYI, here's a table of numbers of such primes less than $10^{k}$:
|N |number of primes |number of primes of the form x^2+y^3 |proportion|
|10 |4 |2 |0.5 |
|100 |25 |8 |0.32 |
|1000 |168 |40 |0.238095 |
|10000 |1229 |192 |0.156225 |
|100000 |9592 |1094 |0.114053 |
|1000000 |78498 |6098 |0.0776835 |
|10000000|665025 |35733 |0.0537318 |
As Elkies said, this result only allows $x, y>0$. If we allow negative $x$ and $y$, thenIf we allow negative $x$ and $y$, then I found an interesting result: every prime $<10^{6}$ can be expressed as $x^{2} + y^{3}$ for some $x, y\in \mathbb{Z}$. However, I don't have any idea to prove this!
EDIT: I found that the above claim is wrong, and there was a problem of overflow. In fact, $p$ is a sum of square and a cube if and only if the elliptic curve $E_{p}:y^{2} = x^{3} +p$ has an interesting result:integral point. Such an elliptic curve is called every primeMordell's curve, and here's $<10^{6}$OEIS link for $n$ such that $E_{n}$ has no integer points. As you can be expressed assee in the list, $x^{2} + y^{3}$$E_{7}$ has no integral solution, i.e. $7 = x^{3} +y^{2}$ has no integer solution. I think this paper might help - which find all the integral points for some $x, y\in \mathbb{Z}$$|n| < 10^{4}$. However, I don't have any idea to prove this!