If $f\in L^p(\mathbb{R}^n)$. Does it make integral sense of type $\int_{\mathbb{R}^n}\mathrm{e}^{ix\cdot \xi}\frac{1}{|\xi|}\widehat{f}(\xi)\,d\xi$? (I ask this because of the possible problem at $\xi=0$)
The reason for my question is due to the fact that the equation
$$ (1-\Delta)u=f$$ has the solution $$u=\mathcal{F}^{-1}\left(\frac{1}{1+|\xi|^2}\widehat{f}(\xi)\right)$$ (or $u(x)=\int_{\mathbb{R}^n}\mathrm{e}^{ix\cdot \xi}\frac{1}{1+|\xi|^2}\widehat{f}(\xi)\,d\xi$)
If $\widehat f\in L^{p_0'} \cap L^{p_1'}$ with $p_0<n<p_1$ (so in dimension $n>1$), then your integral make sense. Writing $$ \int_{\Bbb R^n} \left|\frac{e^{ix\cdot\xi}}{|\xi|}\,\widehat f(\xi)\right|\mathrm d\xi = \int_{|\xi|\leq 1} \frac{|\widehat f(\xi)|}{|\xi|}\mathrm d\xi + \int_{|\xi|\geq 1} \frac{|\widehat f(\xi)|}{|\xi|}\mathrm d\xi $$ and using Hölder's inequality for both integrals yields $$ \int_{\Bbb R^n} \left|\frac{e^{ix\cdot\xi}}{|\xi|}\,\widehat f(\xi)\right|\mathrm d\xi \leq C_0 \,\|\widehat f(\xi)\|_{L^{p_0'}} + C_1 \,\|\widehat f(\xi)\|_{L^{p_1'}} $$ with $C_0 = \left(\int_{|\xi|\leq 1} |\xi|^{-p_0}\,\mathrm d \xi\right)^{1/p_0} < \infty$ if $p_0 <n$, and $C_1 = \left(\int_{|\xi|\leq 1} |\xi|^{-p_1}\,\mathrm d \xi\right)^{1/p_0} < \infty$ if $p_1> n$.
Notice that the singularity at the origin is dealt by the index $p_0$. But if $f\in L^{p_0}$ with $p_0\in[1,2]$, then $\widehat f\in L^{p_0'}$ by the Hausdorff–Young inequality.
So, the true problem is actually the decay of $1/|\xi|$ for large values of $\xi$. This prevents to define your integral as a classical Lebesgue convergent integral.
However, in the distributional sense, there is not problem to define the inverse Fourier transform of $|\xi|^{-1}\,\widehat{f}(\xi)$, and it is nothing but $$ \mathcal F(|\xi|^{-1}\,\widehat{f}(\xi))(x) = C_n\,(-\Delta)^{-1/2}f(x) = c_n\int_{\Bbb R^n} \frac{f(y)}{|x-y|^{n-1}} \mathrm d y. $$ This expression is well defined a.e. if $f\in L^p$ with $p\in(1,n)$ by the Hardy–Littlewood–Sobolev inequality, and more precisely $$ \|(-\Delta)^{-1/2}f\|_{L^q} \leq \|f\|_{L^p} $$ with $q = \frac{np}{n-p}$