Let $\Gamma = \Gamma_1(n)\le\text{SL}_2(\mathbb{Z})$ for some $n$ (might even take $n = 1$).

Let $\mathcal{H}$ be the upper half plane, then on $\mathcal{H}$ we have a line bundle $\mathcal{H}\times\mathbb{C}$. Let $\omega_\Gamma^{an}$ denote the set of isomorphism classes of pairs $(E,P,\omega)$, where $E$ is an elliptic curve over $\mathbb{C}$ and $\omega$ a nonzero holomorphic differential on $E$, and $P$ a point of order $n$.

We may define a map $\mathcal{H}\times\mathbb{C}^\times\rightarrow \omega_\Gamma$ sending $(\tau,t)\mapsto (E_\tau,P_\tau,dz)$ where $E_\tau := \mathbb{C}/\langle 1,\tau\rangle$, $P_\tau := \frac{1}{n} + \langle 1,\tau\rangle$, and $dz$ is a fixed differential on $\mathbb{C}$ pushed forward onto $E_\tau$.

The map $\mathcal{H}\times\mathbb{C}^\times\rightarrow\omega_\Gamma^{an}$ above induces an isomorphism $$(\mathcal{H}\times\mathbb{C}^\times)/\Gamma\cong\omega_\Gamma^{an}$$ where $\Gamma$ acts on $\mathcal{H}\times\mathbb{C}^\times$ via the formula: $$\gamma := \begin{bmatrix} a & b \\ c & d \end{bmatrix}\qquad\gamma(\tau,t) := (\frac{a\tau + b}{c\tau + d},(c\tau + d)t)$$

Forgetting the differential $\omega$ yields a map to the analytic stack $$\omega_\Gamma^{an}\rightarrow[\mathcal{H}/\Gamma]$$ which almost realizes $\omega_\Gamma^{an}$ as a line bundle on $[\mathcal{H}/\Gamma]$, except not quite since the fibers of this map are $\mathbb{C}^\times$. Instead, $\omega_\Gamma^{an}$ sits as an open subspace of the analytic moduli stack $\overline{\omega_\Gamma^{an}}$ of elliptic curves equipped with a differential (possibly zero) and a point of order $n$. Then, the stack $\overline{\omega_\Gamma^{an}}$ should be a line bundle over $[\mathcal{H}/\Gamma]$.

Let $|\overline{\omega_{\Gamma}^{an}}|$ denote the coarse moduli space, then $$|\overline{\omega_{\Gamma}^{an}}| = (\mathcal{H}\times\mathbb{C})/\Gamma$$ and this space admits a natural complex structure given by the projection $\mathcal{H}\times\mathbb{C}\rightarrow(\mathcal{H}\times\mathbb{C})/\Gamma$. There is a standard argument that any holomorphic function $f : \mathcal{H}\rightarrow\mathbb{C}$ satisfying $f(\gamma \tau) = (c\tau + d)^kf(\tau)$ gives rise to a unique holomorphic function $F : |\overline{\omega_\Gamma^{an}}|\rightarrow\mathbb{C}$ satisfying $$F(E,P,\lambda\omega) = \lambda^{-k}F(E,P,\omega),\quad\text{and}\quad F(E_\tau,P_\tau,dz) = f(\tau)$$ In turn, any such a function $F$ can be used to define a section $F' : [\mathcal{H}/\Gamma]\rightarrow (\omega_\Gamma^{an})^{\otimes k}$ by: $$F'(E_\tau,P_\tau) = (E_\tau,P_\tau,F(E_\tau,P_\tau,\omega)\omega^{\otimes k})\qquad \text{$\omega\ne 0$}$$ where the weight-$k$ homogeneity of $F$ means that $F(E_\tau,P_\tau,\omega)\omega^{\otimes k}$ does not depend on the choice of $\omega\ne 0$.

Conversely, given $F'$, one can define $F : |\overline{\omega_\Gamma^{an}}|\rightarrow\mathbb{C}$ via $$F(E_\tau,P_\tau,\omega) = \left\{\begin{array}{ll}F'(E_\tau,P_\tau)/\omega^{\otimes k} & \omega\ne 0 \\ 0 & \omega = 0\end{array}\right.$$ and given $F$, one can recover $f$ by restricting to triples of the form $(E_\tau,P_\tau,dz)$.

Now, algebraically, let $\overline{\omega_\Gamma^{alg}}$ denote the moduli stack (over $\mathbb{C})$ of elliptic curves over $\mathbb{C}$-schemes equipped with a possibly zero differential, and let $\mathcal{M}(\Gamma)$ be the moduli stack over $\mathbb{C}$ of elliptic curves with $\Gamma$-structures.

Thus, $(\overline{\omega_\Gamma^{alg}})^{\otimes k}\rightarrow\mathcal{M}(\Gamma)$ is the algebraic version of $(\overline{\omega_\Gamma^{an}})^{\otimes k}\rightarrow [\mathcal{H}/\Gamma]$. The sections of the former are often called Katz modular forms (over $\mathbb{C}$) of weight $k$ for $\Gamma$, and for now I think I understand how Katz modular forms give rise to sections of the latter, which in turn give rise to holomorphic functions $f : \mathcal{H}\rightarrow\mathbb{C}$ which are weight $k$-invariant under $\Gamma$. By evaluating at the Tate curve, I believe Katz modular forms give rise to weakly holomorphic modular forms. Ie, $f$ is moreover meromorphic at all cusps.

My main question is:

Does every weight $k$ weakly holomorphic modular form $f : \mathcal{H}\rightarrow\mathbb{C}$ (for $\Gamma$), or equivalently a section of $(\overline{\omega_\Gamma^{an}})^{\otimes k}\rightarrow[\mathcal{H}/\Gamma]$ give rise to a Katz modular form? (ie, a section of $(\overline{\omega_\Gamma^{alg}})^{\otimes k}\rightarrow\mathcal{M}(\Gamma))$

If not, at least do the holomorphic modular forms give rise to Katz modular forms? How would we argue this?