A search for the lepton-flavor violating decay of the Higgs boson and potential additional Higgs bosons with a mass in the range 110-160 GeV to an Formula Presented pair is presented. The search is performed with a proton-proton collision dataset at a center-of-mass energy of 13 TeV collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of Formula Presented. No excess is observed for the Higgs boson. The observed (expected) upper limit on the Formula Presented branching fraction for it is determined to be Formula Presented at 95% confidence level, the most stringent limit set thus far from direct searches. The largest excess of events over the expected background in the full mass range of the search is observed at an Formula Presented invariant mass of approximately 146 GeV with a local (global) significance of 3.8 (2.8) standard deviations.