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We critically (re)examine the associated production process of a $J/\psi$meson plus an open charm hadron at the LHC in the proton-proton ($pp$) andproton-lead ($p{\rm Pb}$) collisions. Such a process is very intriguing in thesense of tailoring to explore the double parton structure of nucleons and todetermine the geometry of partons in nuclei. In order to interpret the existing$pp$ data with the LHCb detector at the center-of-mass energy $\sqrt{s}=7$ TeV,we introduce two overlooked mechanisms for the double parton scattering (DPS)and single parton scattering (SPS) processes. Besides the conventional DPSmode, where the two mesons are produced almost independently in the twoseparate scattering subprocesses, we propose a novel DPS mechanism that the twoconstituent (heavy) quarks stemming from two hard scatterings can form into acomposite particle, like the $J/\psi$ meson, during the hadronization phase.However, it turns out the corresponding contribution is small in$J/\psi+c\bar{c}$ hadroproduction. On the contrary, we point out that theresummation of the initial state logarithms due to gluon splitting into a charmquark pair is crucial to understand the LHCb measurement, which was overlookedin the literature. We perform a proper matching between the perturbativecalculations in different initial-quark flavor number schemes, genericallyreferring to the variable flavor number scheme. The new variable flavor numberscheme calculation for the process strongly enhance the SPS cross sections,almost closing the discrepancies between theory and experiment. Finally, wepresent our predictions for the forthcoming LHCb measurement in $p{\rm Pb}$collisions at $\sqrt{s_{NN}}=8.16$ TeV. Some interesting observables areexploited to set up the control regions of the DPS signal and to probe theimpact-parameter dependent parton densities in lead.

$J/\psi$ meson production in association with an open charm hadron at the LHC: A reappraisal