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 |