Sorry to disturb. I recently read Blomer's paper. There is a blur which needs the expert's help here. Blomer's paper says "For any cusp form $f$ of integral weight $k$, level $N$, $f$ can be written as a finite linear combination of Poincare series $P_m$ with $m$ not a square". That is, the Fourier coefficients $a_f(n)$ can be written as

$$a_f(n)=\sum_{\substack{m\ll1\\ m\neq \square}}\alpha_m\sum_{N|c}\frac{S(m,n;c)}{c}J_{k-1} \left( \frac{mn}{c}\right)+O(1)$$for some complex number $\alpha_m$.

Question 1: Since Poincare series $P_m$ can generate a cuspidal form, whether or not we have a quantitative bound for $m$? For example $m\ll |S_k(N)|\sim kN$.

Question 2: For a half-weight cusp form $g\in S_{l+\frac{1}{2}}(N)$, whether we can simply write the Fourier coefficients $a_g(n)$ of $g$ as the Fourier coefficients of one or two Poincare series $𝑃_m$, with the same wight $l+\frac{1}{2}$?

Sorry if there is something inaccurate about description of these two questions. Wish some expert here can share his/her comments. Thanks in advance.