It follows from so-called global central limit theorems (see e.g. Corollary 1, part 1)) that $$\bigg|E\Big|\frac Xn-p\Big|-\sqrt\frac2\pi\,\sqrt{\frac{pq}n}\bigg|<\frac Cn$$ for some absolute real constant and all natural $n$.

It follows from so-called global central limit theorems (see e.g. Corollary 1, part 1), on page 2 of "Download preview PDF.") that $$\bigg|E\Big|\frac Xn-p\Big|-\sqrt\frac2\pi\,\sqrt{\frac{pq}n}\bigg|<\frac Cn$$ for some absolute real constant $C$ and all natural $n$.