  We discuss soft-gluon resummation for the associated production of a single
top quark and a Higgs boson ($tqH$ production) in single-particle-inclusive
(1PI) kinematics. We present analytical results for the higher-order
corrections and numerical results for the cross sections at LHC energies. We
calculate approximate NNLO total rates, including scale dependence and
uncertainties from parton distributions, as well as top-quark
transverse-momentum and rapidity distributions. In all cases we find that the
soft-gluon corrections are dominant and provide important contributions to the
total and differential cross sections.
