An inclusive search for anomalous Higgs boson production in the diphoton decay channel and in association with at least one jet is presented, using LHC proton-proton collision data collected by the CMS experiment at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The razor variables $M_\mathrm{R}$ and $\mathrm{R}^2$, as well as the momentum and mass resolution of the diphoton system, are used to categorize events into different search regions. The search result is interpreted in the context of strong and electroweak production of supersymmetric particles. We exclude bottom squark pair-production with masses below 450 GeV for bottom squarks decaying to a bottom quark, a Higgs boson, and the lightest supersymmetric particle (LSP) for LSP masses below 250 GeV. For wino-like chargino-neutralino production, we exclude charginos with mass below 170 GeV for LSP masses below 25 GeV. In the GMSB scenario, we exclude charginos with mass below 205 GeV for neutralinos decaying to a Higgs boson and a goldstino LSP with 100% branching fraction.