I am looking for a cone $X\subseteq\mathbb A^n_{\mathbb C}$ (defined by homogeneous equations) and an irreducible homogeneous polynomial $f$ in $n$ variables such that $U := D(f)\cap X = X_f = \{ x\in X \mid f(x)\ne 0\}$ is smooth and $f$ has a $k$-th root $g\in \mathcal O_X(U)$ (for some $k>1$), but it has no root in $\mathcal O_X(X)$. Clearly, $X$ can not be a normal variety: If it were, the fact that $g$ does not have a pole at the divisor $Z(f)\cap X$ would imply that it extends to $\mathcal O_X(X)$. However, we are looking for an example where this is not the case.
I actually know an example where this happens, but it is in high dimension and sometimes intuitively hard to grasp - particularly the non-normality often leads me astray. I would like to find a toy example with similar properties in small dimensions, something like $n\le 3$ would be the best. I have toyed around with the "usual" examples of singular curves and surfaces but could not construct $f$ with all the right properties.
PS: Feel free to work projectively, my formulation comes from the fact that in concrete examples, I personally find it easier to think of cones.