  We present the detection and follow-up observations of planetary candidates
around low-mass stars observed by the K2 mission. Based on light-curve
analysis, adaptive-optics imaging, and optical spectroscopy at low and high
resolution (including radial velocity measurements), we validate 16 planets
around 12 low-mass stars observed during K2 campaigns 5-10. Among the 16
planets, 12 are newly validated, with orbital periods ranging from 0.96-33
days. For one of the planets (K2-151b) we present ground-based transit
photometry, allowing us to refine the ephemerides. Combining our K2 M-dwarf
planets together with the validated or confirmed planets found previously, we
investigate the dependence of planet radius $R_p$ on stellar insolation and
metallicity [Fe/H]. We confirm that for periods $P\lesssim 2$ days, planets
with a radius $R_p\gtrsim 2\,R_\oplus$ are less common than planets with a
radius between 1-2$\,R_\oplus$. We also see a hint of the "radius valley"
between 1.5 and 2$\,R_\oplus$ that has been seen for close-in planets around
FGK stars. These features in the radius/period distribution could be attributed
to photoevaporation of planetary envelopes by high-energy photons from the host
star, as they have for FGK stars. For the M dwarfs, though, the features are
not as well defined, and we cannot rule out other explanations such as
atmospheric loss from internal planetary heat sources, or truncation of the
protoplanetary disk. There also appears to be a relation between planet size
and metallicity: those few planets larger than about 3 $R_\oplus$ are found
around the most metal-rich M dwarfs.
