Hardy-Littlewood first conjecture asserts that the number $\pi_{n}(x)$ of prime gaps of size $n$ below $x$ is asymptotic to $2C_{n}\dfrac{x}{\log^2 x}$, where $C_n$ depends only on $n$.
Has it been proven that there exist two functions $f$ and $h$ such that $\pi_{n}(x)=f(n)h(x)+A_{n}(x)\sim f(n)h(x)$? If so, writing $x+O(1)=\sum_{n=2}^{G(x)}n(f(n)h(x)+A_{n}(x))$, where $G(x)$ is the maximal prime gap below $x$, would it be possible to get an unconditional upper bound for $G(x)$ in terms of $h(x)$?
My idea is that the different error terms $A_{n}(x)$ cancel out up to a bounded quantity when summed over $n$.