Using $(2.712\pm 0.014)\times {10}^9$ $\psi (3686)$ events collected by the BESIII detector at the BEPCII collider, the $\psi (3686)\to \gamma p\bar{p}{\pi }^{+}{\pi }^{-}{\pi }^0$ process is investigated. Evidence for the decay of ${\eta }_c(2S)\to p\bar{p}{\pi }^{+}{\pi }^{-}{\pi }^0$ is found with a signal significance of 3.3$\sigma$. The product of branching fractions of $ℬ[\psi (3686)\to \gamma {\eta }_c(2S)]\times ℬ[{\eta }_c(2S)\to p\bar{p}{\pi }^{+}{\pi }^{-}{\pi }^0]$ is determined to be $(3.4\pm 0.5\pm 0.8)\times {10}^{-6}$, where the first uncertainty is statistical and the second systematic. The hadronic decays of ${\chi }_{cJ}\to p\bar{p}{\pi }^{+}{\pi }^{-}{\pi }^0$$(J=0,1,2)$ are observed, and their branching fractions are measured to be $ℬ({\chi }_{c0}\to p\bar{p}{\pi }^{+}{\pi }^{-}{\pi }^0)=(4.79\pm 0.01\pm 0.40)\times {10}^{-3}$, $ℬ({\chi }_{c1}\to p\bar{p}{\pi }^{+}{\pi }^{-}{\pi }^0)=(2.13\pm 0.01\pm 0.17)\times {10}^{-3}$, and $ℬ({\chi }_{c2}\to p\bar{p}{\pi }^{+}{\pi }^{-}{\pi }^0)=(3.72\pm 0.01\pm 0.29)\times {10}^{-3}$, respectively. Furthermore, the branching fractions for the intermediate processes ${\chi }_{cJ}\to p\bar{p}\omega$ are updated with significantly improved precision: $ℬ({\chi }_{c0}\to p\bar{p}\omega )=(5.76\pm 0.01\pm 0.42)\times {10}^{-4}$, $ℬ({\chi }_{c1}\to p\bar{p}\omega )=(1.85\pm 0.01\pm 0.13)\times {10}^{-4}$, and $ℬ({\chi }_{c2}\to p\bar{p}\omega )=(4.51\pm 0.01\pm 0.33)\times {10}^{-4}$, respectively.