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The process $\psi(3686) \to \pi^0 h_c, h_c \to \gamma \eta_c$ has beenstudied with a data sample of $106 \pm 4$ million $\psi(3686)$ events collectedwith the BESIII detector at the BEPCII storage ring. The mass and width of the$P$-wave charmonium spin-singlet state $h_c(^1P_1)$ are determined bysimultaneously fitting distributions of the $\pi^0$ recoil mass for 16exclusive $\eta_c$ decay modes. The results, $M(\hc) = 3525.31 \pm 0.11 {\rm(stat.)} \pm 0.14 {\rm (syst.)}$\,MeV/$c^2$ and $\Gamma(\hc) = 0.70 \pm 0.28\pm 0.22$\,MeV, are consistent with and more precise than previousmeasurements. We also determine the branching ratios for the 16 exclusive$\eta_c$ decay modes, five of which have not been measured previously. Newmeasurements of the $\eta_c$ line-shape parameters in the $E1$ transition$h_c\to\gamma\eta_c$ are made by selecting candidates in the $h_c$ signalsample and simultaneously fitting the hadronic mass spectra for the 16 $\eta_c$decay channels. The resulting $\eta_c$ mass and width values are $M(\eta_c) =2984.49 \pm 1.16 \pm 0.52$\,MeV/$c^2$ and $\Gamma(\eta_c) = 36.4 \pm 3.2 \pm1.7$\,MeV.

Study of $\psi(3686)\to\pi^0 h_c, h_c\to\gamma\eta_c$ via $\eta_c$ exclusive decays