In Hardy-Littlewood's 1923 paper *"Some problems of 'Partitio Numerorum' III"* it is proven, assuming a weak version of GRH (namely that there is $\varepsilon>0$ s.t. all zeroes of $L(s,\chi)$ have $\Re(s)<3/4-\varepsilon$), that for all $k\geqslant 3$, when $n\to\infty$ through the integers with same parity than $k$:
$$ r_{k}(n) \sim \frac{2C_k}{(k-1)!}\frac{n^{k-1}}{\log(n)^k}\prod_{\substack{p\mid n \\ p\geqslant 3}} \left(\frac{(p-1)^k+(-1)^k(p-1)}{(p-1)^k -(-1)^k} \right), $$

where $r_{k}(n):= \{(p_1,\ldots,p_k)\in\mathbb{P}^k:\sum p_i = n \}$ and $C_k$ is a constant given by: $$ C_k := \prod_{p\geqslant 3} \left(1- \frac{(-1)^k}{(p-1)^k}\right). $$

Roughly 15 years later Vinogradov introduced his influencial technique that allowed him to prove this estimate unconditionally for $k=3$. One thing that bothers me is: *what about the other $k$?*

The wikipedia article on Goldbach's conjecture has (as of today [08/Oct/2016]) the following passage:

This formula has been rigorously proven to be asymptotically valid for $k \geqslant 3$ from the work of Vinogradov, but is still only a conjecture when $k=2$.$^{\text{[citation needed]}}$

I gave a quick look at K. F. Roth & Anne Davenport's translation of Vinogradov's The Method of Trigonometrical Sums in the Theory of Numbers, and in Chapter X. "Goldbach's Problem" it has the following passage at the start:

In the present chapter I give a solution of Goldbach's problem concerning the representability of every sufficiently large odd number $N$ as the sum of three primes, and I establish an asymptotic formula for the number of representations.

The method used here enables one also to solve more general additive problems involving primes, for example the question of representability of large numbers $N$ in the form $$ N = p_1^n + \ldots + p_s^n $$ (Waring's problem for primes). But I do not consider these more general questions here.

Every other source I consulted only talks about the case $k=3$. Does Vinogradov's method for $k=3$ implies the other cases as a corollary? Where can I find more (historical) information about the other $k$ in this Hardy & Littlewood's estimate? Thanks in advance!