If $p > n/2$ and if $f \in C^2_c (\mathbb{R}^n \setminus \{0\})$ (twice continuously differentiable functions whose support is compact in $\mathbb{R}^n \setminus \{0\}$), then the weighted Hardy inequality to $f$ says that 
$$
  \int_{\mathbb{R}^n} \frac{| f (x) |^p}{| x |^{2 p}} \,dx
\le \Bigl(\frac{p}{2 p - n}\Bigr)^p \int_{\mathbb{R}^n} \frac{| \nabla f (x) |^p}{| x |^{p}} \,dx
$$
Next an application of the classical Hardy inequality gives you, if $p \ne n$, 
$$
  \int_{\mathbb{R}^n} \frac{| \nabla f (x) |^p}{| x |^{p}} \,dx
\le \Bigl(\frac{p}{p - n}\Bigr)^p \int_{\mathbb{R}^n} |D^2 f (x) |^p \,dx.
$$

If $p \in (n/2, n)$, then such functions are dense in the subspace of $W^{2, p} (\mathbb{R}^n)$ of functions that vanish at $0$, and this proves the desired inequality.

If $p > n$, then the inequality  $W^{2, p} (\mathbb{R}^n)$ of functions that vanish at $0$ **together with their derivative**. It is important that the derivative vanishes as it can be checked that if $f \in C^2 (\mathbb{R}^n)$, $f = 0$then 
$$
  \frac{| f (x) |^p}{| x |^2 p} \simeq \frac{\rvert \nabla f (0)\lvert}{|x|^p},
$$
which is not integrable near the origin when $p > n$.