Let $\Omega := B_{\frac{1}{2}}(0) \subseteq \mathbb{R}^n$ and $n>1$. It is rather well known that $u: \Omega \setminus \lbrace 0 \rbrace \rightarrow \mathbb{R}$ lives in the Sobolev space $W^{1, n}(\Omega)$, where $$ u(x) := \log \lvert \log \lvert x \rvert \rvert. $$ This is in spite of the fact that $u$ can not be extended such that it is a function in $C^1(\Omega)$. An approximation argument can be applied to show weak differentiability while using that $u \in C^1(\Omega \setminus \lbrace 0 \rbrace)$.
Now, out of pure interest: Do you know an example of a function $f \in C^1(\Omega \setminus \lbrace 0 \rbrace)$ with $f, \nabla f \in L^p(\Omega)$ (for some $1 \leq p \leq \infty$) but that is not weakly differentiable, i.e. $f \notin W^{1, p}(\Omega)$?
When you write $\nabla f\in L^p(\Omega)$ I have to interpret it as the statement:
Then, in one dimension, consider on $\Omega\setminus\{0\}:=(-1,1)\setminus\{0\}$ the indicator function $\chi$ of $(0,1)$. Clearly $\chi$ is in $L^\infty(-1,1)$ as well as its classical gradient (which is identically zero). But the distributional derivative is a measure (Dirac measure). Therefore it is not in any of the Sobolev spaces $W^{1,p}(-1,1)$.
You can cook up similar examples in any dimension if you increase the dimension of the singular set. In fact, you are essentially asking when $W^{1,p}(\Omega)$ and $W^{1,p}(\Omega\setminus \{0\})$ are different. This should happen only in dimension one as the question is related to the capacity of a set (the capacity of the singleton $\{0\}$ here).