  The vector-tensor Horndeski theory is supplemented by a real, massless scalar
field non-minimally coupled to the Horndeski interaction term. The generic
dyonic Reissner-Nordstrom solutions characterized by electric and magnetic
charges, acquire an instability and bifurcate into electromagnetically charged
solutions presenting an horizon and a cloud of scalar hairs at specific values
of the non-minimal coupling constant. These hairy black holes are studied in
terms of the physical parameters of the model and the critical phenomena
limiting their domain of existence are discussed.
