Odd-dimensional Weyl and pseudo-Weyl spaces admitting almost contact, almost paracontact and nilpotent structures are considered in this work. The results are obtained by means of the apparatus of the prolonged covariant differentiation. A linear connection with torsion is constructed. With respect to this connection the prolonged covariant derivatives of the fundamental tensors of the Weyl and pseudo-Weyl spaces are found to be zero. The curvature tensor with respect to this connection is considered.