  We obtain asymptotics of large Hankel determinants whose weight depends on a
one-cut regular potential and any number of Fisher-Hartwig singularities. This
generalises two results: 1) a result of Berestycki, Webb and Wong [5] for
root-type singularities, and 2) a result of Its and Krasovsky [37] for a
Gaussian weight with a single jump-type singularity. We show that when we apply
a piecewise constant thinning on the eigenvalues of a random Hermitian matrix
drawn from a one-cut regular ensemble, the gap probability in the thinned
spectrum, as well as correlations of the characteristic polynomial of the
associated conditional point process, can be expressed in terms of these
determinants.
