  Data in $B$-meson decays indicate violations of lepton flavour universality,
thereby raising the question about such phenomena in the charm sector. We
perform a model-independent analysis of NP contributions in (semi)-leptonic
decays of $D_{(s)}$ mesons which originate from $c \to d \bar{\ell} \nu_{\ell}$
and $c \to s \bar{\ell} \nu_{\ell}$ charge-current interactions. Starting from
the most general low-energy effective Hamiltonian containing four-fermion
operators and the corresponding short-distance coefficients, we explore the
impact of new (pseudo)-scalar, vector and tensor operators and constrain their
effects through the interplay with current data. We pay special attention to
the elements $|V_{cd}|$ and $|V_{cs}|$ of the Cabibbo--Kobayashi--Maskawa
matrix and extract them from the $D_{(s)}$ decays in the presence of possible
NP decay contributions, comparing them with determinations utilizing unitarity.
We find a picture in agreement with the Standard Model within the current
uncertainties. Using the results from our analysis, we make also predictions
for leptonic $D_{(s)}^+ \to e^+ \nu_e$ modes which could be hugely enhanced
with respect to their tiny Standard Model branching ratios. It will be
interesting to apply our strategy at the future high-precision frontier.
