Let $D(\epsilon,C)$ be the collection of all random variables $X$ on $\mathbb{R}$ such that $E[X]=0$, $E[X^2]=1$, and $E[|X|^{2+\epsilon}]\leq C$. Define a function $L\_{\epsilon,C}(n)$ by
$$L\_{\epsilon,C}(n) = \sup\_{X\in D(\epsilon,C)} \sup\_{x\in\mathbb{R}} \left\vert P\left(\frac{X\_1+X\_2+\ldots+X\_n}{\sqrt{n}}<x\right)-P\left(Z < x\right)\right\vert$$
where $X\_i$ are independent copies of $X$, and $Z$ is a standard normal.


Having made these definitions, the Berry-Esseen theorem tells us that
$$L\_{1,C}(n) = O(C n^{-1/2})$$
It is also not hard to use the Lindeberg-Feller theorem to show that for any fixed $\epsilon>0$ and $C\in \mathbb{R}$
$$\lim\_{n\to\infty} L\_{\epsilon,C}(n) = 0$$
Which leads to my question -- can we get any more quantitative result than that this goes to zero, without assuming $\epsilon=1$? Or, in the other direction, is it known that we can't?



Berry-Esseen type bounds with lower moment assumptions
$\newcommand{\ep}{\epsilon}$


To answer your question, one can use e.g. a result by [Bikelis](https://books.google.com/books?id=zSDqCAAAQBAJ&q=bikelis#v=snippet&q=bikelis&f=false), which states the following: Let $X\_1,\dots,X\_n$ be independent zero-mean random variables such that $E|X\_i|^{2+\ep}<\infty$ for some $\epsilon\in[0,1]$ and all $i$. Then for all real $x$ 
\begin{equation}
 \Big|P\Big(\sum\_1^n X\_i<x\sqrt B\_n\Big)-P(Z<x)\Big|
 \le \frac A{B\_n^{1+\ep/2}(1+|x|)^{2+\ep}}\,\sum\_1^n E|X\_i|^{2+\ep}, 
\end{equation}
where $A$ is a finite universal constant, $B\_n:=\sum\_1^n EX\_i^2$, and $Z$ is a standard normal random variable. 


In you specific case, it follows that $L\_{\epsilon,C}\le AC/n^{\epsilon/2}$. 


