  We consider the series expansion of the $L^p$-Hardy inequality of
\cite{BFT2}, in the particular case where the distance is taken from an
interior point of a bounded domain in $\mathbb{R}^n$ and $1<p\neq n$. For $p<n$
we improve it by adding as a remainder term an optimally weighted critical
Sobolev norm, generalizing the $p=2$ result of \cite{FT} and settling the open
question raised in \cite{BFT1}. For $p>n$ we improve it by adding as a
remainder term the optimally weighted H\"{o}lder seminorm, extending the
Hardy-Morrey inequality of \cite{Ps} to the series case.
