Measurements of $CP$ observables in $B\pm \to D(\ast )K\pm$ and $B\pm \to D(\ast )\pi \pm$ decays are presented, where $D(\ast )$ indicates a neutral $D$ or $D\ast$ meson that is an admixture of meson and anti-meson states. Decays of the $D\ast$ meson to the $D\pi 0$ and $D\gamma$ final states are partially reconstructed without inclusion of the neutral pion or photon. Decays of the $D$ meson are reconstructed in the $K\pm \pi \mp$, $K+K-$, and $\pi +\pi -$ final states. The analysis uses a sample of charged $B$ mesons produced in proton-proton collisions and collected with the LHCb experiment, corresponding to integrated luminosities of 2.0, 1.0, and 5.7 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ taken at centre-of-mass energies of 7, 8, and 13 TeV, respectively. The measurements of partially reconstructed $B\pm \to D\ast K\pm$ and $B\pm \to D\ast \pi \pm$ with $D\to K\mp \pi \pm$ decays are the first of their kind, and a first observation of the $B\pm \to (D\pi 0)D\ast \pi \pm$ decay is made with a significance of 6.1 standard deviations. All $CP$ observables are measured with world-best precision, and in combination with other LHCb results will provide strong constraints on the CKM angle $\gamma$.