  We discuss charged and static solutions in a shift-symmetric scalar-tensor
gravity model including a negative cosmological constant. The solutions are
only approximately Anti-de Sitter (AdS) asymptotically. While spherically
symmetric black holes with scalar-tensor hair do exist in our model, the
uncharged spherically symmetric scalar-tensor solitons constructed recently
cannot be generalised to include charge. We point out that this is due to the
divergence of the electric monopole at the origin of the coordinate system,
while higher order multipoles are well-behaved. We also demonstrate that black
holes with scalar hair exist only for horizon value larger than that of the
corresponding {\it extremal} Reissner-Nordstr\"om-AdS (RNAdS) solution, i.e.
that we cannot construct solutions with arbitrarily small horizon radius. We
demonstrate that for fixed $Q$ a horizon radius exists at which the specific
heat $C_Q$ diverges - signalling a transition from thermodynamically unstable
to stable black holes. In contrast to the RNAdS case, however, we have only
been able to construct a stable phase of large horizon black holes, while a
stable phase of small horizon black holes does not (seem to) exist.
