For the convenience of the reader, I have written Hardy and Littlewood's conjecture from their paper linked in the comments above:
Suppose that $a,b,c$ are integers and $a$ is positive; that $\gcd(a,b,c) = 1$; that $a+b$ and $c$ are not both even; and that $D = b^2 - 4ac$ is not a square. Then there are infinitely many primes of the form $am^2 + bm + c$. The number $P(n)$ of such primes less than $n$ is given asymptotically by
$$\displaystyle P(n) \sim \frac{\varepsilon C}{\sqrt{a}} \frac{\sqrt{n}}{\log n} \prod_{p | \gcd(a,b)} \left(\frac{p-1}{p} \right) $$
where $\varepsilon$ is $1$ if $a + b$ is odd and $2$ if $a+b$ is even. The constant $C$ is given by
$$\displaystyle C = \prod_{w \geq 3, w \nmid a} \left(1 - \frac{1}{w-1} \left(\frac{D}{w} \right)\right).$$
To answer the question, there has not been any recent work on this problem for some time due to its perceived difficulty. In fact it is a long-standing debate between prominent analytic number theorists whether this conjecture is harder or easier than the twin prime conjecture. Despite the significant progress made by Zhang and Maynard about ten years ago on the latter, we are still very far away from twin primes themselves.
The best results towards this conjecture are due to Iwaniec (Almost primes represented by quadratic polynomials), who proved that the polynomial $x^2 + 1$ takes on infinitely many integer values with at most two prime divisors (including multiplicity) and that the density of such values is $\gg \sqrt{n}/\log(n)$. His arguments can be readily generalized to arbitrary quadratic polynomials satisfying the conditions imposed by Hardy and Littlewood above. On the other hand, Iwaniec's arguments have no chance of being improved to give actual primes without very significant new ideas, due to the parity barrier.
