  Observations of X-ray binaries indicate a dearth of compact objects in the
mass range from $\sim 2-5 M_{\odot}$. The existence of this (first mass) gap
has been used to discriminate between proposed engines behind core-collapse
supernovae. From LIGO/Virgo observations of binary compact remnant masses,
several candidate first mass gap objects (either neutron stars (NSs) or black
holes (BHs)) were identified during the O3 science run. Motivated by these new
observations, we study the formation of BH-NS mergers in the framework of
isolated classical binary evolution, using population synthesis methods to
evolve large populations of binary stars (Population I and II) across cosmic
time. We present results on the NS to BH mass ratios ($q=M_{\rm NS}/M_{\rm
BH}$) in merging systems, showing that although systems with a mass ratio as
low as $q=0.02$ can exist, typically BH-NS systems form with moderate mass
ratios $q=0.1-0.2$. If we adopt a delayed supernova engine, we conclude that
$\sim 30\%$ of BH-NS mergers may host at least one compact object in the first
mass gap (FMG$^\circ$). Even allowing for uncertainties in the processes behind
compact object formation, we expect the fraction of BH-NS systems ejecting mass
during the merger to be small (from $\sim 0.6-9\%$). In our reference model, we
assume: (i) the formation of compact objects within the FMG, (ii) natal NS/BH
kicks decreased by fallback, (iii) low BH spins due to Tayler-Spruit angular
momentum transport in massive stars. We find that $\lesssim 1\%$ of BH-NS
mergers will have any mass ejection and about the same percentage will produce
kilonova bright enough to have a chance of being detected with a large
(Subaru-class) $8$m telescope. Interestingly, all these mergers will have both
a BH and an NS in the FMG.
