  The flavor-changing neutral current (FCNC) decays of charmed hadrons with
missing energy $(\not\!\!E)$ can serve as potentially promising hunting grounds
for hints of new physics, as the standard-model backgrounds are very
suppressed. A few of such processes have been searched for in recent
experiments, specifically $D^0\to\,\not\!\!E$ by Belle and
$D^0\to\pi^0$$\not\!\!E$ and $\Lambda_c^+\to p\!\not\!\!E$ by BESIII, resulting
in upper bounds on their branching fractions. We consider them to illuminate
the possible contributions of the quark transition $c\to u\!\not\!\!E$ with a
couple of invisible spinless bosons carrying away the missing energy, assuming
that they are not charge conjugates of each other and hence can have unequal
masses. We find that these data are complementary in that they constrain
different sets of the underlying operators and do not cover the same ranges of
the bosons' masses, but there are regions not yet accessible. From the allowed
parameter space, we show that other $D$-meson decays, such as
$D\to\rho$$\not\!\!E$, and the charmed-baryon ones
$\Xi_c\to(\Sigma,\Lambda)$$\not\!\!E$ can have sizable branching fractions and
therefore may offer further probes of the new-physics interactions. We point
out the importance of $D^0\to\gamma\!\not\!\!E$ which are not yet searched for
but could access parts of the parameter space beyond the reach of the other
modes. In addition, we look at a scenario where the invisibles are instead
fermionic, namely sterile neutrinos, and a scalar leptoquark mediates $c\to
u\!\not\!\!E$. We discuss the implications of the aforesaid bounds for this
model. The predictions we make for the various charmed-hadron decays in the
different scenarios may be testable in the near future by BESIII and Belle II.
