Prime values of f(a,b2) and f(a,p2), f quadratic

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 of 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.