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!