ByBerry-Esseen theorem on Wikipedia we know that $$|F_n(x)-\Phi(x)|\le \frac{C\rho}{\sigma^3\sqrt{n}}$$ where $F_n$ is the cumulative distribution function given there.
However, in many important cases we expect $F_n(0)$ to be much closer or equal to $\Phi(0).$ For example if $p=1/2$ and $n$ is odd, then $$F_n(0)=\sum_{k=0}^{\lfloor n/2\rfloor} {n\choose k} p^k(1-p)^{n-k}=\frac{1}{2}=\Phi(0).$$
(By $F_n(0)$ above, I really mean we consider a slightly modified binomial distribution, but I hope this is clear.) Is there a better bound for $|F_n(x)-\Phi(x)|$ for the example above in terms of a function $E(x)$ that goes to zero as $x\to 0$ and achieves a maximum that is less than or equal to $\frac{C\rho}{\sigma^3\sqrt{n}}$? Is there a more general error term $E(x)$ that works for other binomial distributions?
For a sequence of zero-mean i.i.d. r.v.s $\{X_i\}$ with variance $\sigma^2$ and finite third moment $\gamma_3$, $$ |\mathsf{P}(S_n/(\sqrt{n}\sigma)\le x)-\Phi(x)|\le \frac{C\gamma_3}{\sigma^3\sqrt{n}}\times \frac{1}{1+|x|^3}, $$ where $S_n:=\sum_{i\le n}X_i$ and $C>0$ is an absolute constant (see, e.g. Chen and Shao, 2001). This bound is better then the uniform one for large values of $x$. However, if the distribution of $S_n$ is known, you may get better estimates.