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.of the Schwarzschild black hole characterized by a new parameter : the so-called NUT charge n. Although purely analytic, the NUT space-time presents peculiarities [16,17] that makes that its physical interpretation is, till now, a matter of debate. In particular, the solution presents a Misner string singularity on the polar axis and the corresponding space-time contains closed timelike curves. Various arguments rehabilitating space-time with a NUT charge are proposed in [18]. In spite of the difficulty of finding a global definition of the NUT space-time, the solution possesses many remarquable properties, namely : (i) like the Kerr solution, it is stationary but non-static due to non-vanishing g tϕ metric terms ; (ii) it can be extended analytically (i.e. without curvature singularity) in the interior region by means of a TAUB solution.Likely for these reasons, several authors (see namely [19,20]) have considered the NUT parameter as a possible ingredient of some astrophysical object and have studied its effect on geodesics in NUT space-times. Another application of the NUT parameter was proposed recently in [21] to obtain families of non-trivial, spherically symmetric solutions of the Einstein-Chern-Simons gravity coupled to a scalar field. Such a construction was possible by taking advantage of the stationary character of the underlying metric.In this paper we extend the construction of the hairy black holes of [10] by including a NUT parameter in the metric. We show that Nutty-hairy-black holes exist in a specific domain of the NUT charge and Gauss-Bonnet parameter. A special emphasis is set on the way the NUT charge affects the solution in the interior of the black hole. Also, we study the influence on the light-like geodesic of both the presence of the scalar field and of the NUT charge. It is found in particular that, mimicking a rotation, the NUT charge leads to a non-planar drift of the trajectories.The paper is organized as follows. In Sect. 2 we present the model, the ansatz for the metric, the boundary conditions of the ensuing classical equations and sketch the form of a perturbative solution. The non-perturbative solutions, obtained with a numerical method, are reported in Sect. 3. The influence of the Gauss-Bonnet gravity term and of the NUT parameter on the light-like geodesics are emphasized in Sect. 4 and illustrated by some figures. Conclusion and perspectives are given in Sect. 5.