I know that $\frac{1}{x \pm i0}$ is a tempered distribution in $\mathcal{S}'(\mathbb{R})$, see e.g. the Sokhotski–Plemelj theorem. In some lecture notes online I found the following statement (without proof):
If $H:\mathbb{R}^n \to \mathbb{R}$ and $|\nabla H| \neq 0$ at any point where $H(x)=0$ then $\frac{1}{H(x) \pm i0}$ is in $\mathcal{S}'(\mathbb{R}^n).$
We can also assume that $H$ "behaves nice" and is a $C^\infty$-function, so let's say $H$ is a polynomial, e.g. $H(x_1, \dots, x_n)=x_1^2+ \dots +x_n^2-1.$ So far I could not find any proof for this statement but it should be true since similiar distributions are used in PDE theory.
Remark: $\frac{1}{H(x) + i0}$ is defined as $\phi \mapsto \lim_{\varepsilon \to 0}\int_{\mathbb{R}^n}\frac{\phi(x)}{H(x) + i\varepsilon}dx$.