  We construct a one-parameter family of solutions to the positive singular
Q-curvature problem on compact nondegenerate manifolds of dimension bigger than
four with finitely many punctures. If the dimension is at least eight we assume
that the Weyl tensor vanishes to sufficiently high order at the singular
points. On a technical level, we use perturbation methods and gluing techniques
based on the mapping properties of the linearized operator both in a small ball
around each singular point and in its exterior. Main difficulties in our
construction include controlling the convergence rate of the Paneitz operator
to the flat bi-Laplacian in conformal normal coordinates and matching the
Cauchy data of the interior and exterior solutions; the latter difficulty
arises from the lack of geometric Jacobi fields in the kernel of the linearized
operator. We overcome both these difficulties by constructing suitable
auxiliary functions.
