  In this paper, we investigate the shadows and rings of the charged Horndeski
black hole illuminated by accretion flow that is both geometrically and
optically thin. We consider two types of accretion models: spherical and
thin-disk accretion flow. We find that in both types of models, the size of the
charged Horndeski black hole shadow decreases with the increase of the charge,
and it decreases more slowly for the Reissner-Nordstr\"om (RN) black hole. In
the spherical accretion flow model, we find that the increase of the charge of
Horndeski black hole brightens the light ring around it, and it brightens more
significantly in comparison with RN black hole. Due to the Doppler effect, the
charged Horndeski black holes with accretion flow of radial motion have darker
shadows than those with the static accretion flow, but the size of the shadow
is not affected by accretion flow motion. In the thin disk-shaped accretion
flow model, we find that the brightness of the light ring around the charged
Horndeski black hole is dominated by the direct emission from the accretion
flow, and the contribution from lensed rings is relatively small, and that from
the photon rings is negligible. We also find that the ring brightness decreases
as the charge of Horndeski black hole increases, and the decrease is more
significant than that in the RN black hole case. Moreover, the radiation
position of the accretion flow can affect the shadow size and the ring
brightness of the charged Horndeski black hole.
