  We apply a recently developed 2+1+1 decomposition of spacetime, based on a
nonorthogonal double foliation for the study of spherically symmetric, static
black hole solutions of Horndeski scalar-tensor theory. Our discussion proceeds
in an effective field theory (EFT) of modified gravity approach, with the
action depending on metric and embedding scalars adapted to the nonorthogonal
2+1+1 decomposition. We prove that the most generic class of Horndeski
Lagrangians compatible with observations can be expressed in this EFT form. By
studying the first order perturbation of the EFT action we derive three
equations of motion, which reduce to those derived earlier in an orthogonal
2+1+1 decomposition, and a fourth equation for the metric parameter N related
to the nonorthogonality of the foliation. For the Horndeski class of theories
with vanishing $G_3$ and $G_5$, but generic functions $G_2(\phi,X)$ (k-essence)
and $G_4(\phi)$ (nonminimal coupling to the metric) we prove the unicity
theorem that no action beyond Einstein--Hilbert allows for the Schwarzschild
solution. Next we integrate the EFT field equations for the case with only one
independent metric function obtaining new solutions characterized by a
parameter interpreted as either mass or tidal charge, the cosmological constant
and a third parameter. These solutions represent naked singularities, black
holes with scalar hair or have the double horizon structure of the
Schwarzschild--de Sitter spacetime. Solutions with homogeneous Kantowski--Sachs
type regions also emerge. Finally, one of the solutions obtained for the
function $G_4$ linear in the curvature coordinate, in certain parameter range
exhibits an intriguing logarithmic singularity lying outside the horizon. The
newly derived hairy black hole solutions evade previously known unicity
theorems by being asymptotically nonflat, even in the absence of the
cosmological constant.
