  We thoroughly investigate both transverse momentum and threshold resummation
effects on scalar-pseudoscalar pair production via quark-antiquark annihilation
at the $13~ \text{TeV}$ Large Hadron Collider at QCD NLO+NLL accuracy. A
factorization method is introduced to properly supplement the soft-gluon
(threshold) resummation contribution from parton distribution functions to the
resummed results obtained by the Collins-Soper-Sterman resummation approach. We
find that the impact of the threshold-resummation improved PDFs is comparable
to the resummation effect of the partonic matrix element and can even
predominate in high invariant mass regions. Moreover, the loop-induced
gluon-gluon fusion channel in the type-I two-Higgs-doublet model is considered
in our calculation. The numerical results show that the electroweak production
via quark-antiquark annihilation dominates over the gluon-initiated QCD
production by $1 \sim 2$ orders of magnitude.
