  The 2HDM+$a$ model is one of the main models used in the interpretations of
dark matter searches at the LHC. So far, all the 2HDM+$a$ benchmarks considered
by the ATLAS and CMS experiments are limited to a type-II Yukawa sector, in
which the Higgs bosons $A$, $H$, and $H^\pm$ are all constrained to be
mass-degenerate and heavier than around $600 \, {\rm GeV}$. In this work, we
present the first detailed study of 2HDM+$a$ models with a type-I Yukawa
sector, which, for moderate values of $\tan\beta$, lift the constraints from
flavour physics, allowing the extra Higgs bosons to be even lighter than the
$125 \, {\rm GeV}$ Higgs boson discovered at the LHC. We discuss several
benchmarks where the $A$, $H$, and $H^\pm$ states are not necessarily
mass-degenerate and the signatures that arise in these models, some of which
have not yet been explored at the LHC. We present the dominant channels in the
studied benchmarks and the expected sensitivity in Run 2 data using truth-level
analyses and discuss potential improvements in the experimental searches for
Run 3.
