Given a Siegel modular form $F = F(\tau, z, \sigma)$ of weight $k$ and degree 2 with respect to $Sp_{4}(\mathbb{Z})$, we have the Fourier-Jacobi expansion:
$$F(\tau, z, \sigma) = \sum_{m=0}^{\infty} \varphi_{k,m}(\tau, z) p^{m},$$
where $p=e^{2 \pi i \sigma}$ and $\varphi_{k,m}$ is a holomorphic Jacobi form of weight $k$ and index $m$. Is there a theory or collection of examples where holomorphic Jacobi forms are replaced by weak Jacobi forms? I have an inkling it should give rise to meromorphic Siegel modular forms perhaps. But the literature seems remarkably sparse here, unless I am looking in the wrong places.
(One avenue I understand and am not particularly interested in for this question is the relationship between the elliptic genus of K3 (a weak Jacobi form of weight 0 and index 1) with the Igusa cusp form $\chi_{10}(\tau, z, \sigma)$. I would be more interested in examples involving $\varphi_{-2,1}$ the unique weak Jacobi form of weight -2 and index 1.)