In what paper do Hardy and Littlewood first mention, specifically, their 2nd conjecture? It is not mentioned specifically in Partitio Numerorum III.
This conjecture is usually expressed as
$$\pi(x+y)\leq \pi(x)+\pi(y). $$
in which $\pi(x)$ is the number of primes not exceeding $x.$
See, for example, Mathworld's note.
Hardy-Littlewood's second conjecture does appear in the paper you cite.$^*$
The article is 70 pages long and the idea is briefly noted at pp. 52-54. The article is cited for Hardy-Littlewood's second conjecture in dozens of places and fortunately one gave the pages.
At page 52 the authors introduce a difference.
"The general case raises very interesting questions as to the density of the distribution of primes, and it will be convenient, to begin by discussing them. We write
(5.6II) $$\rho(x) = \overline{lim}_{n\to \infty}(\pi(n+x)-\pi(n)) $$
so that $\rho(x)= \rho([x])$ is the greatest number of primes that occurs indefinitely often in a sequence $n+1,n+2,...,n+[x]$ of $[x]$ consecutive integers."
And on page 54 they conjecture:
"An examination of the primes less than 200 suggests forcibly that
$$\rho(x) \leq \pi(x),~~(x\geq 2). "$$
But although the methods we are about to explain lead to striking conjectural lower bounds, they throw no light on the problem of an upper bound."
At page 68, after related digression, the authors give a calculation and re-state their sense that the conjecture appears plausible.
Beyond $x = 97$ it would seem that $\rho(x)$ falls further below $\pi(x)$, at least within any range in which calculation is practicable.
This conjecture, unlike quite a few others, is not named in the paper as far as I can tell. The usual statement of Hardy-Littlewood's second conjecture is
$$\pi(n+x) \leq \pi(n)+\pi(x).$$
$^*$ G.H Hardy and J.E. Littlewood, Some problems of ‘partitio numerorum:’ III: On the expression of a number as a sum of primes, Acta Mathematica, December 1923, Volume 44, pp.1-70. The full text of the paper is available via Springer. There is a paywall so I cannot link to it. The first five pages are available free at several sites.
This is the answer I posted to a similar question some time ago at MathOverflow.
I took a look at Schinzel and Sierpinski, Sur certaines hypotheses concernant les nombres premiers, Acta Arith IV (1958) 185-208, reprinted in Volume 2 of Schinzel's Selecta, pages 1113-1133. In the Selecta, there is a commentary by Jerzy Kaczorowski, who mentions "the G H Hardy and J E Littlewood conjecture implicitly formulated in [33] that $\pi(x+y)\le\pi(x)+\pi(y)$ for $x,y\ge2$." [33] is Partitio Numerorum III. Schinzel and Sierpinski (page 1127 of the Selecta) define $\rho(x)=\limsup_{y\to\infty}[\pi(y+x)-\pi(y)]$, and point to that H-L paper, pp 52-68. They then write (page 1131), "$\bf C_{12.2}.$ L'hypothese de Hardy et Littlewood suivant laquelle $\rho(x)\le\pi(x)$ pour $x$ naturels $\gt1$ equivaut a l'inegalite $\pi(x+y)\le\pi(x)+\pi(y)$ pour $x\gt1,y\gt1$." It should be said that the proof that the first inequality implies the second relies on Hypothesis H, which essentially says that if there is no simple reason why a bunch of polynomials can't all be prime, then they are, infinitely often.
Schinzel and Sierpinski express no opinion as to any degree of belief in the conjecture under discussion.
I don't suppose this actually answers any of the questions, although Kaczorowski's use of the word "implicitly" may be significant.
