generalized functions & operators

We can define the Laplacian of such functions $f$ as a signed measure (that is, a measure that can assign positive or negative values to sets). Precisely, the generalized/distributional Laplacian $\Delta f$ is a measure $\mu$ such that for any smooth compactly support function $\varphi$ we have $$\int f\, \Delta\varphi =\int \varphi\,d\mu \tag1$$

When $f$ is twice differentiable, $\mu$ is just a continuous measure with density $\Delta f$. In general, it may have singular part. In your example (assuming you work in $\mathbb R^3$) one can calculate $\Delta |x|^{-1}$ by applying Green's identity in a spherical shell $r<|x|<R$ and letting $r\to 0$. Choose $R$ large so that the support of $\varphi$ is within $|x|<R$. The normal derivative in direction away from the origin is denoted by $\frac{\partial }{\partial n} $ below.

$$\begin{split} \int_{r <|x|<R} |x|^{-1}\, \Delta\varphi(x) &= \int_{r <|x|<R} (|x|^{-1}\, \Delta\varphi(x) - \varphi(x)\Delta(|x|^{-1})) \\ &= \int_{|x|=r} \left( \frac{\partial |x|^{-1} }{\partial n} \varphi(x) - \frac{\partial \varphi }{\partial n} |x|^{-1} \right) \\ & =-r^{-2}\int_{|x|=r} |x|^{-2} \varphi(x) + r^{-1}\int_{|x|=r} \frac{\partial \varphi }{\partial n} \\ &\to -4\pi \varphi(0) +0 = \int \varphi\,d\mu \end{split} \tag2$$ where $\mu$ is $-4\pi\delta_0$.

A useful theorem: $\Delta f$ is a nonnegative measure if and only if $f$ is subharmonic. In this example $\Delta_f$ is nonpositive, hence $f$ is superharmonic.

Since the Laplacian is a linear operator, you can deal with your function (which apparently has other things besides $1/r$ by writing it $f(x)=1/|x|+g(x)$ where $g$ may be sufficiently nice to have Laplacian in the classical sense.