I want to use holomorphic projection to derive some formulas for the Fourier-coefficients of different smooth functions that transform with a holomorphic integer weight $k$, that is $f: \mathbb{H} \mapsto \mathbb{C}$ with \begin{align*} f(\gamma \tau) = (c \tau +d) ^{k} f(\tau) \end{align*} for $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_{2}(\mathbb{Z})$ and $\tau \in \mathbb{H}$.
In "Projections of $C^{\infty}$ automorphic forms" Sturm described the holomorphic projection, which was later generalized by Zagier in "Introduction to modular forms" for functions with $f(\tau)=c_0 + O(v^{-\epsilon})$ as $v=Im(\tau) \rightarrow \infty$ for some $\epsilon >0$. Zagiers projection "only" yields modular forms of weight $k$.
In "Formulas for non-holomorphic Eisenstein series and for the Riemann zeta function at odd integers" O´Sullivan mentions that for functions that have at most polynomial growth at $i\infty$, one can define a projection that yields a cusps form of weight "k", but does not give any references for this. Any references to sources where such a projection is described would be appreciated.