  We discover a non-trivial rather simple relation between the mock modular
generating functions of the level $1$ and level $N$ Hurwitz class numbers. This
relation yields a holomorphic modular form of weight $\frac{3}{2}$ and level
$4N$, where $N > 1$ is stipulated to be odd and square-free. We extend this
observation to a non-holomorphic framework and obtain a higher level
non-holomorphic Zagier Eisenstein series as well as a higher level preimage of
it under the Bruinier--Funke operator $\xi_{\frac{1}{2}}$. All of these
observations are deduced from a more general inspection of a certain weight
$\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point
$s=\frac{3}{4}$. This idea goes back to Duke, Imamo\={g}lu and T\'{o}th in
level $4$ and relies on the theory of so-called sesquiharmonic Maass forms. We
calculate the Fourier expansion of our sesquiharmonic preimage and of its
shadow. We conclude by offering an example if $N=5$ or $N=7$ and we provide the
SAGE code to compute the Fourier coefficients involved.
