Let me try to shed some light on this question by providing a summation formula for the modified sum $$S(X) = \sum_{n\in\mathbb{Z}} \frac{A_F(n,1)}{\vert n\vert}e\left(n\frac{a}{c}\right)W\left(\frac{n}{X}\right),$$ for a smooth function $W$ with compact support in $\mathbb{R}_+$. To derive this formula it will be useful to switch to the language of automorphic representations. So let us assume that $f$ and $F$ are newforms and let $\pi_f$ and $\Pi_F=\text{sym}^2(\pi_f)$ be the associated automorphic representations.
By applying Theorem~1.1 from A. Corbett's paper "Voronoi summation for $\text{GL}_n$: collusion between level and modulus" with $\chi=1$, $c_2=1$, $\phi_{\infty}=W(\frac{\cdot}{X})$, $M=1$, $l=1$ and $q=c$ we get $$S(X)=c\sum_{\substack{m,r\in\mathbb{Z}_{\neq 0},\\ (m,N)=1,\\ r\mid N^{\infty}}}\sum_{d\mid c}\text{KL}(\overline{aN^5}r;m;c,1,d)\frac{A_F(d,m)}{\vert md\vert}\mathcal{B}_{\Pi_{F,\infty}, \phi_{\infty}}\left(\frac{rmd^2}{c^3N^5}\right)\prod_{p\mid N} \mathcal{B}_{\Pi_{F,p}}\left(\frac{rmd^2}{c^3N^5}\right).$$ Note that in the special case where $f$ has trivial nebentypus and squarefree level $N$ the conductor of $\Pi_F$ turns out to be $N^2$. Above $\text{KL}$ is a 2-dimensional Kloosterman sum defined by $$\text{KL}(x;y;c,1,d) = \sum_{\alpha\in \left(\mathbb{Z}/\frac{c}{d}\mathbb{Z}\right)^{\times}}e\left(-\frac{xd\alpha}{c}+\frac{y\overline{\alpha}}{c/d}\right)$$ and the functions $\mathcal{B}_{*,*}(\cdot)$ are certain Hankel-transforms.
One might argue if this is already a Voronoi-summation formula or something else. Unfortunately the right hand side is not explicit enough for many purposes in analytic number theory. Therefore let me describe some of the elements in more detail. First of all note that the Fourier coefficients $A_F(m,d)$ depend only on unramified data of $\pi_F$ and $\mathcal{B}_{\Pi_{F,\infty}, \phi_{\infty}}$ depends only on the archimedean type of $\pi_f$. This is well understood in the literature see for exapmle Proposition~5.12 in "Summation formulas, from Poisson and Voronoi to the present" and also the paper A. Corbett mentioned above.
The $p$-adic Hankel-transforms $\mathcal{B}_{\Pi_{F,p}}$ are much harder to deal with in general as these are weighted sums of twisted $\epsilon$-factors. Therefore in general one might try to work with soft bounds and support properties. However in the special situation at hand there is some hope. Indeed, fix $p\mid N$ and observe that $\pi_{f,p}$ can only be the Steinberg/special representation and $\Pi_{F,p}$ can be computed rather explicitly. The functions $\mathcal{B}_{\Pi_{F,p}}$ can then be evaluated using equation (27) in A. Corbetts paper as above, where one takes $\Phi$ to be the suitable Euler-factor coming form $A_F(p^r,1)$.
I hope this is of some use and I apologise for not giving more details in the last paragraph.