A 1997 paper of Étienne Fouvry and Henryk Iwaniec, Gaussian primes, concerns the prevalence of primes that are of the form $n^2+p^2$ for prime $p$. The asymptotic result is
$\sum_{n^2+p^2\le x}\Lambda(p)\Lambda(n^2+p^2)=kx+O(x(\log x)^{-A})$
with $A>0$ arbitrary and $k=2\prod_{p>2}\left(1-\frac{\chi(p)}{(p-1)(p-\chi(p))}\right)\approx2.1564103447695$ where $\chi(n)=(-1)^{(p-1)/2}$ is the nontrivial character mod 4. The big-O constant is uniform, depending only on the choice of $A$.
I would like to use this to find an asymptotic formula for $f(x):=|\mathcal{P}\cap\{n^2+p^2\le x\}|$. It looks like
$f(x)=2k\frac{x}{(\log x)^2}(1+o(1))$
but I'm not quite sure of my derivation, nor even of how to interpret the original result (are duplicate representations double-counted or not?). Can someone confirm or deny my calculation?
Bonus question: were Fouvry & Iwaniec the first to show that there are infinitely many of these primes? They cite Rieger, Coleman, Duke, and Pomykala as related results but none had both prime restrictions.