Let $p$ be prime. For an integer weight $k$ modular form $f$ with $p$-integral coefficients, we have the filtration modulo $p$ defined as $$w_{p}(f) := \min\{k' : \overline{f} = \overline{g}, \quad g \text{ has weight $k$}\},$$ where $\overline{f}$ is the $q$-series obtained by reducing the coefficients of $f$ modulo $p$. This gives a notion of "weight" for mod $p$ modular forms. One then typically analyzes filtrations of derivatives $\theta := q\frac{d}{dq}$ of $f$ (which are modular forms mod $p$). Noting that $\overline{\theta^{p}(f)} = \overline{\theta(f)}$, we obtain the finite sequence $$w_{p}(f), w_{p}(\theta(f)), \ldots, w_{p}(\theta^{p-1}(f)),$$ which I will call the theta cycle of $f$. The theta cycle gives information about Ramanujan congruences, for example, among other things.
Question: I am wondering if there is a similar theory of mod $p$ modular forms when $k \in \frac{1}{2}\mathbb{N}$? The above theory hinges on the grading $$M(SL_{2}(\mathbb{Z})) = \bigoplus_{k \in \mathbb{N}} M_{k}(SL_{2}(\mathbb{Z})),$$ the left-hand side being the space of all modular forms (of level 1) and the right a direct sum over all possible weights. When reducing mod $p$ we almost get the same grading, but up to multiples of $p-1$ due to the fact that the Eisenstein series $\overline{E}_{p-1} = \overline{1}$.
Is there a way to talk about the notion of "weight" for reductions (mod $p$) of half-integer weight modular forms? Is there a nice version of $\theta$ for such forms? I would like to be pointed to any literature on the subject.