  We study odd parity perturbations of spherically symmetric black holes with
time-dependent scalar hair in shift-symmetric higher-order scalar-tensor
theories. The analysis is performed in a general way without assuming the
degeneracy conditions. Nevertheless, we end up with second-order equations for
a single master variable, similarly to cosmological tensor modes. We thus
identify the general form of the quadratic Lagrangian for the odd parity
perturbations, leading to a generalization of the Regge-Wheeler equation. We
also investigate the structure of the effective metric for the master variable
and refine the stability conditions. As an application of our generalized
Regge-Wheeler equation, we compute the quasi-normal modes of a certain
nontrivial black hole solution. Finally, our result is extended to include the
matter energy-momentum tensor as a source term.
