Consider $X_m$ which are independent, identically distributed random variables which have moments exactly up to order $2$ and no higher. This can be done in numerous ways. One is the following: let $a_n$ be a sequence of positive numbers that goes to zero and let $b_n$ be a sequence of positive numbers such that $\sum_{n=1}^\infty \frac{b_n}{a_n}$ converges. (For example, $a_n=1/n,b_n=2^{-n}$ does the job.) Let $f_n(x)=(2+a_n) 1_{[1,\infty)}(x) |x|^{-3-a_n}$. Then let $f(x)=\sum_{n=1}^\infty \frac{b_n}{\sum_{k=1}^\infty b_k} f_n(x)$. Then $f$ is a pdf of a r.v. which has no higher moments than the second (since each $f_n$ has exactly moments of order strictly less than $2+a_n$). The assumption about $b_n$ ensures that it actually does have a finite second moment. (In effect the trick here is $\bigcap_{n=1}^\infty [1,2+a_n)=[1,2]$.)
Despite this somewhat pathological moment property, the classical central limit theorem still tells us that $\frac{\overline{X}_m-\mu}{\sigma/\sqrt{m}}$ converges in distribution to a $N(0,1)$ random variable. However, the most common quantitative estimates for the convergence rate, e.g. the Berry-Esseen theorem, require the existence of a moment higher than order $2$. What can be said about the convergence rate in cases like the one in the previous paragraph?
The paper
Friedman, N.; Katz, Melvin; Koopmans, L. H. Convergence rates for the central limit theorem. Proc. Nat. Acad. Sci. U.S.A. 56 1966 1062–1065.,
available here, seems to give something in this direction. The authors prove that for an i.i.d. sequence of centered random variables with unit variance and such that $\mathbb E\left[X_1^2\log\left(1+\left|X_1\right|\right)\right]< +\infty$, the series $$\sum_{n=1}^{+\infty } \frac 1n\left|\mathbb P\left\{ \frac{S_n } { \sqrt n }\leqslant x\right\} - \Phi(x) \right| $$ is convergent for any $x\in\mathbb R\setminus \{0\}$, where $S_n=\sum_{i=1}^nX_i$ and $\Phi$ denotes the cumulative distribution function of a standard normal distribution.
However, we still have a little bit more than a finite second moment.