  Einstein gravity supplemented by a scalar field non-minimally coupled to a
Gauss-Bonnet term provides an example of model of scalar-tensor gravity where
hairy black holes do exist. We consider the classical equations within a metric
endowed with a NUT-charge and obtain a two-parameter family of nutty-hairy
black holes. The pattern of these solutions in the exterior and the interior of
their horizon is studied in some details. The influence of both -- the hairs
and the NUT-charge -- on the lightlike and timelike geodesics is emphasized.
