  In this paper, we compute a low order approximation of a system of large
order $n$ that matches $\nu$ moments of order $j_i$ of the transfer function,
at $\nu$ interpolation points, has $\ell$ poles and $k$ zeros fixed and also
matches $\nu-(\ell +k)$ moments of order $j_i+1$, where $j_i+1$ is the
multiplicity of the $i$-th interpolation point. We derive explicit linear
systems in the free parameters to simultaneously achieve the required pole-zero
placement and match the desired high order moments. We compute the closed form
of the free parameters that meet the constraints, as the solution of a $\nu$
order linear system. Furthermore, for data-driven model reduction, we
generalize the construction of the Loewner matrices to include the data and the
imposed pole and higher order moment constraints. The resulting approximations
achieve a trade-off between the good norm approximation and the preservation of
the dynamics of the original system in a region of interest.
