  Current experimental information on the charm meson decay observables in
which the $c\to s\ell\nu_\ell$ transitions occur is well compatible with the
Standard Model predictions. Recent precise lattice calculations of the $D_s$
meson decay constant and form factors in $D\to K\ell\nu$ decays offer a
possibility to search for the small deviations from the Standard Model
predictions in the next generation of the high intensity flavour experiments.
We revisit constraints from these processes on the new physics contributions in
the effective theory approach. We investigate new physics effects which might
appear in the differential distributions for the longitudinally and
transversely polarised $K^\ast$ in $D\to K^\ast\ell\nu_\ell$ decays. Present
constraints from these observables are rather weak, but could be used to
constrain new physics effects in the future. In the case of $D\to K\ell\nu$ we
identify observables sensitive on new physics contribution coming from the
scalar Wilson coefficient, namely the forward-backward and the transversal muon
asymmetries. By allowing that new physics modifies only the second lepton
generation but not the first one, we identify allowed region for the
differential decay rate for the process $D\to K\mu\nu_\mu$ and find that it is
allowed to deviate from the Standard Model prediction by only few percent. The
lepton flavour universality violation can be tested in the ratio
$R_{\mu/e}(q^2)\equiv \frac{d\Gamma^{(\mu)}}{dq^2}/\frac{d\Gamma^{(e)}}{dq^2}$.
If the first lepton generation behaves as in the Standard Model, we find, using
current constraint on the scalar Wilson coefficient, that the ratio
$R_{\mu/e}(q^2)$ is currently allowed to be within the range $(0.9, 1.2)$,
depending on the value of $q^2$.
