A search is presented for electroweak production of supersymmetric particles in final states with one lepton, a Higgs boson decaying to a pair of bottom quarks, and large missing transverse momentum. The search uses data from proton-proton collisions at a center-of-mass energy of 13~\mathrm{TeV} collected with the CMS detector at the CERN LHC, corresponding to an integrated luminosity of 137~\mathrm{fb}^{-1}. The observed data yields are consistent with the estimated standard model backgrounds. Exclusions are set in the context of a simplified supersymmetric model of chargino-neutralino production, with the chargino decaying to a W boson and the lightest supersymmetric particle (LSP) and the neutralino decaying to a Higgs boson plus an LSP. Charginos and neutralinos with masses up to 820~\mathrm{GeV} are excluded at $95\%$ confidence level when the LSP mass is small, and LSPs with mass up to 350~\mathrm{GeV} are excluded when the mass of the chargino and neutralino is about 700~\mathrm{GeV}.

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