  We prove an asymptotic formula for primes of the shape $f(a,b^2)$ with $a,b$
integers and of the shape $f(a,p^2)$ with $p$ prime. Here $f$ is a binary
quadratic form with integer coefficients, irreducible over $\mathbb{Q}$ and has
no local obstructions. This refines the seminal work of Friedlander and Iwaniec
on primes of the form $x^2 + y^4$ and Heath-Brown and Li on primes of the form
$a^2 + p^4$, as well as earlier work of the author with Lam and Schindler on
primes of the form $f(a,p)$ with $f$ a positive definite form.
