I present a consistent way to include $\eta$-$\eta^\prime$
mixing in global analyses of two-body decays of heavy hadrons employing the approximate flavour-SU(3) symmetry of QCD. The framework is applied to $D\to P \eta^\prime$ decays, where $P$ denotes a pseudoscalar meson. The result shows that flavour-SU(3) symmetry holds in the decay rates of these modes to better than 30\%. With future data we expect the branching ratios of $D_s\to K^+ \eta^\prime$ and $D \to K^+\eta^\prime$ to move upward and downward by $\sim\!\! 1\sigma$, respectively. Subsequently I discuss the implications of the LHCb measurements of the CP asymmetries in $D\to K^+K^-$ and $D\to \pi^+\pi^-$ for generic scenarios of new physics. New-physics contributions should have imprints on other CP asymmetries as well and can be tested through sum rules. Promising decays are $D_s^+\to K^0\pi^+$, $D^+\to \bar K^0K^+$, $D^0\to K^0 \bar K^{*0}$, $D^0\to \bar K^0 K^{*0}$, $D_s^+\to K^{*0}\pi^+$, and
$D^+\to \bar K^{*0}K^+$.