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

We prove an asymptotic formula for primes of the shape f(a,b²) with $a,b$ integers and of the shape f(a,p²) with p prime. Here f is a binary quadratic form with integer coefficients, irreducible over Q and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form x² + y⁴ and Heath-Brown and Li on primes of the form a² + p⁴, 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.