  Static spherically symmetric black holes are discussed in the framework of
higher dimensional gravity with quadratic in curvature terms. Such terms
naturally arise as a result of quantum corrections induced by quantum fields
propagating in the gravitational background. We focus our attention on the
correction of the form ${\cal C}^2=C_{\alpha\beta\gamma\delta}
C^{\alpha\beta\gamma\delta}$. The Gauss-Bonnet equation in four-dimensional
(4D) spacetime enables one to reduce this term in the action to the terms
quadratic in the Ricci tensor and scalar curvature. As a result the
Schwarzschild solution which is Ricci flat will be also a solution of the
theory with the Weyl scalar ${\cal C}^2$ correction. An important new feature
of the spaces with dimension $D > 4$ is that in the presence of the Weyl
curvature-squared term a solution necessary differs from the corresponding
`classical' vacuum Tangherlini metric. This difference is related to the
presence of {\em secondary} or {\em induced} hair. We explore how the
Tangherlini solution is modified by `quantum corrections', assuming that the
gravitational radius $r_0$ is much larger than the scale of the quantum
corrections. We also demonstrated that finding a general solution beyond the
perturbation method can be reduced to solving a single third order ODE (master
equation).
