A search for a light charged Higgs boson ($\mathrm{H}^{+}$) decaying to a W boson and a CP-odd Higgs boson ($\mathrm{A}$) is performed using data from proton-proton collisions at \sqrt{s}=13~\mathrm{TeV}. The data were recorded by the CMS detector at the LHC in 2016 and correspond to an integrated luminosity of 35.9~\mathrm{fb}^{-1}. In this search, it is assumed that the $\mathrm{H}^{+}$ boson is produced in decays of top quarks, and the $\mathrm{A}$ boson decays to two opposite-charge muons. The presence of signals for $\mathrm{H}^{+}$ boson masses between $100$ and 160~\mathrm{GeV} and $\mathrm{A}$ boson masses between $15$ and 75~\mathrm{GeV} is investigated. No evidence for the production of the $\mathrm{H}^{+}$ boson is found. Assuming branching ratios $\mathcal{B}(\mathrm{H}^{+}\to\mathrm{W}^{+}\mathrm{A})=1$ and $\mathcal{B}(\mathrm{A}\to\mu^{+}\mu^{-})=3\times10^{-4}$, upper limits at $95\%$ confidence level on the branching fraction of the top quark, $\mathcal{B}(\mathrm{t}\to\mathrm{b}\mathrm{H}^{+})$, of $0.63$ to $2.9\%$ are obtained, depending on the masses of the $\mathrm{H}^{+}$ and $\mathrm{A}$ bosons.

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