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 $y$, 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!