  We study $n\times n$ Hankel determinants constructed with moments of a
Hermite weight with a Fisher-Hartwig singularity on the real line. We consider
the case when the singularity is in the bulk and is both of root-type and
jump-type. We obtain large $n$ asymptotics for these Hankel determinants, and
we observe a critical transition when the size of the jumps varies with $n$.
These determinants arise in the thinning of the generalised Gaussian unitary
ensembles and in the construction of special function solutions of the
Painlev\'e IV equation.
