We obtain large $N$ asymptotics for $N \times N$ Hankel determinants corresponding to non-negative symbols with Fisher-Hartwig (FH) singularities in the multi-cut regime. Our result includes the explicit computation of the multiplicative constant. More precisely, we consider symbols of the form $ωe^{f-NV}$, where $V$ is a real-analytic potential whose equilibrium measure $μ_V$ is supported on several intervals, $f$ is analytic in a neighborhood of $\textrm{supp}(μ_V)$, and $ω$ is a function with any number of jump- and root-type singularities in the interior of $\textrm{supp}(μ_V)$. While the special cases $ω\equiv1$ and $ωe^f\equiv1$ have been considered previously in the literature, we also prove new results for these special cases. No prior asymptotics were available in the literature for symbols with FH singularities in the multi-cut setting. As an application of our results, we discuss a connection between the spectral fluctuations of random Hermitian matrices in the multi-cut regime and the Gaussian free field on the Riemann surface associated to $μ_V$. As a second application, we obtain new rigidity estimates for random Hermitian matrices in the multi-cut regime.