Going from $\Lambda$ to a prime count

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.

Concerning the bonus question, the review in Math Reviews says the authors prove this result, and the reviewer doesn't mention anyone else having done it. I would take this as evidence that Fouvry and Iwaniec were the first.