As pointed out in the comments such a formula is not explicitly in the literature. However, it can be derived from a general Voronoi type formula due to A. Corbett. Let me try to shed some light on how to do this in the following. For simplicity I will treat 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.

Before we can come to the summation formula in question we gather some preliminaries. Note that, because $N$ is square free and $f$ has trivial nebentypus, we know that $\pi_{f,p}=\text{St}$ is the Steinberg representation for all $p\mid N$. From this we can derive several useful facts. At the archimedean place we find that $\Pi_{F,\infty}$ has representation parameters $$\lambda=(2\nu,-2\nu,0) \text{ and } \delta=\begin{cases} (0,0,0) &\text{ in the Maaß case,}\\ (1,0,1) &\text{in the holomorphic case.}\end{cases} $$ In particular, $$ L_{\infty}(s,\Pi_{F}) = \begin{cases}
	\Gamma_{\mathbb{R}}(s+2\nu)\Gamma_{\mathbb{R}}(s)\Gamma_{\mathbb{R}}(s-2\nu) &\text{ in the Maaß case,} \\ \Gamma_{\mathbb{C}}(s+2k-1)\Gamma_{\mathbb{R}}(s+1) &\text{ in the holomorphic case.}
\end{cases}$$ See for example Proposition~5.12 in ["Summation formulas, from Poisson and Voronoi to the present"](https://arxiv.org/abs/math/0304187). The associated gamma-quotient then is $$\gamma(s,\Pi_{F,\infty},\psi_{\infty})
= \epsilon(s,\Pi_{F,\infty},\psi_{\infty})\frac{L_{\infty}(1-s,\tilde{\Pi}_{F})}{L_{\infty}(s,\Pi_{F})}.$$
The computation at the finite places is slightly more involved. We start by mentioning that the Steinberg representation corresponds under the Local Langlands Correspondence to $\Vert \cdot \Vert^{-\frac{1}{2}}\otimes \text{sp}(2)$, where $\text{sp}(2)$ is the $2$-dimensional special Deligne-representation of $W_{\mathbb{Q}_p}$. Further, using Artin reciprocity we can identify a character $\chi$ of $\mathbb{Q}_p^{\times}$ with a character of $W_{\mathbb{Q}_p}$. The upshot is that, we can compute the $L$- and $\epsilon$-factors of $\chi \Pi_{F,p}$ on the Weil-Deligne side. Indeed we find that 
\begin{equation}
	\left[\Vert \cdot \Vert^{-\frac{1}{2}}\otimes \text{sp}(2)\right]\otimes \left[\Vert \cdot \Vert^{-\frac{1}{2}}\otimes\text{sp}(2)\right] \cong 1 \oplus \left[ \Vert \cdot \Vert^{-1}\otimes\text{sp}(3)\right]. \nonumber
\end{equation}
From this we conclude that instead of computing the local factors of $\chi \Pi_{F,p}$ we can compute those of $\chi\Vert \cdot \Vert^{-1}\otimes \text{sp}(3)$ on the Weil-Deligne side of the Local Langlands Correspondence. According to ["Elliptic Curves and the Weil-Deligne group"](https://math.berkeley.edu/~dyott/Elliptic%20Curves%20and%20the%20Weil-Deligne%20Group.pdf) by D. E. Rohrlich we have that 
\begin{multline}
	a(\chi \Pi_{F,p}) = \begin{cases}
		2 &\text{ if $\chi$ is unramified,}\\
		3a(\chi) &\text{ if $\chi$ is ramified,}
	\end{cases}, L(s,\Pi_{F,p}) = L(s+1,\chi) \\ \text{ and } \epsilon(\frac{1}{2},\chi\Pi_{F,p}) = \begin{cases}
		1 &\text{ if $\chi=1$,}\\ \epsilon(\frac{1}{2},\chi)^3 &\text{ if $\chi$ is unitary and ramified.}
 	\end{cases} \nonumber
\end{multline} 
(Here some clarification concerning the epsilon factors is necessary. We are always following Langlands convention and consider the canonical additive character with the corresponding self dual Haar measure as fixed and drop them from the notation.)

Our local computation implies the well known fact that the conductor of $\Pi_F$ is $N^2$. Further, we can write
\begin{multline}
	A_F(m,n)=A_F\left(\frac{m}{(m,N^{\infty})},\frac{n}{(n,N^{\infty})}\right) (mn,N^{\infty})\\ \cdot \prod_{p\mid N} W_{F,p}\left(\left(\begin{matrix} (mn,p^{\infty})&0&0\\0&(n,p^{\infty})&0\\0&0&1\end{matrix}\right)\right), \nonumber 
\end{multline} 
where $W_{F,p}$ denotes the (suitably normalised) Whittaker new vector of $\Pi_{F,p}$. The Fourier coefficients $A_F(m,d)$ depend only on unramified data of $\pi_F$. In particular we have the following well known relations:
\begin{equation*}
	A_F(m,1) = A_F(1,m) = \sum_{d^2\mid n} a_f(\frac{n^2}{d^4}) \text{ and } A_F(m,n) = \sum_{d\mid (m,n)}\mu(d)A_F(\frac{n}{d},1)A_F(\frac{n}{d},1),
\end{equation*}
for $(md,N)=1$. Knowing the local $L$-factor we can compute
\begin{multline}
	W_{F,p}\left(\left(\begin{matrix} p^{f_1+f_2} & 0 & 0\\ 0 & p^{f_2} & 0 \\ 0 & 0 &1\end{matrix}\right)\right) = p^{-f_1-f_2}s_{(f_1+f_2,f_2)}(p^{-1},0)W_{F,p}(1) \\= \begin{cases} p^{-2f_1}W_{F,p}(1) &\text{ if }f_2=0, \\
	0& \text{ if } f_2\neq 0.
\end{cases} \nonumber
\end{multline}
This can be deduced from Theorem 4.1 in ["Whittaker functions associated to newforms of GL(n)"](https://projecteuclid.org/euclid.jmsj/1390600834) by M. Miyauchi.

We now make the first step towards our summation formula. By applying Theorem~1.1 from A. Corbett's paper ["Voronoi summation for $\text{GL}_n$: collusion between level and modulus"](https://arxiv.org/abs/1807.00716) with $\chi=1$, $c_2=1$, $\phi_{\infty}=W(\frac{\cdot}{X})$, $M=1$, $l=1$, $q=c$ and $\phi_p=1_{\mathbb{Z}_p}$ for $p\mid N$ we get $$S(X)=c\sum_{\substack{m,r\in\mathbb{Z}_{\neq 0},\\ (m,N)=1,\\ r\mid N^{\infty}}}\sum_{d\mid c}\text{KL}(\overline{aN^5}r;m;c,1,d)\frac{A_F(d,m)}{\vert md\vert}\mathcal{B}_{\Pi_{F,\infty}, \phi_{\infty}}\left(\frac{rmd^2}{c^3N^5}\right)\prod_{p\mid N} \mathcal{B}_{\Pi_{F,p}}\left(\frac{rmd^2}{c^3N^5}\right).$$ Here $\text{KL}$ is a 2-dimensional Kloosterman sum defined by $$\text{KL}_2(x,y;\frac{c}{d})\colon =\text{KL}(x;y;c,1,d) = \sum_{\alpha\in \left(\mathbb{Z}/\frac{c}{d}\mathbb{Z}\right)^{\times}}e\left(-\frac{xd\alpha}{c}+\frac{y\overline{\alpha}}{c/d}\right)$$ and the function $\mathcal{B}_{\Pi_{F,\infty},\phi_{\infty}\infty}(\cdot)$ is a Bessel-transform of $\phi_{\infty}$ given by
$$\mathcal{B}_{\Pi_{F,\infty},\phi_{\infty}\infty}(y) = \frac{1}{4\pi i}\sum_{r=0,1} \text{sgn}(y)^r\int_{(\sigma)}\gamma(1-s,\text{sgn}^r\Pi_{F,\infty},\psi_{\infty})\vert y\vert^{1-s}\int_0^{\infty}W(\frac{x}{X})\vert x\vert^{-1-s}dxds.$$
The transforms $\mathcal{B}_{\Pi_{F,p},\phi_{p}}(\cdot)$ are $p$-adic versions of this Bessel transform. They are given by
\begin{multline}
\mathcal{B}_{\Pi_{F,p},\phi_{p}}(y)= \frac{\log(p)}{2\pi} \sum_{\substack{\xi\colon F^{\times}\to S^1,\\ \xi(p)=1}}\xi(y)\int_{\sigma-i\frac{\pi}{\log(p)}}^{\sigma+i\frac{\pi}{\log(p)}}\epsilon(\frac{1}{2},\xi\Pi_{F,p})p^{a(\xi \Pi_{F,p})(s-\frac{1}{2})}\vert y \vert_p^{1-s} \\ \cdot \int_{\mathbb{Q}_p^{\times}} \xi(x)\phi_p(x)W_{F,p}\left(\left(\begin{matrix}
 x&0&0\\0&1&0\\0&0&1\end{matrix}\right)\right)\vert x\vert_p^{-s}d^{\times}x ds. \nonumber
\end{multline}
We see that in our particular case the $x$-integral vanishes for all non-trivial characters. A short computation shows
$$\mathcal{B}_{\Pi_{F,p},\phi_{p}}(y)= \frac{\log(p)}{2\pi}\int_{\sigma-i\frac{\pi}{\log(p)}}^{\sigma+i\frac{\pi}{\log(p)}} L(s,\Pi_{F,p})p^{2s-1}\vert y \vert_p^{1-s}ds = \delta_{v_p(y)\geq -2}p^{-3-2v_p(y)}.  $$

In summary we have seen that
$$S(X)=\frac{c}{N^3}\sum_{m\in\mathbb{Z}_{\neq 0}}\sum_{d\mid c}\text{KL}_2(\overline{aN^2},m;\frac{c}{d})\frac{A_F(m,d)}{\vert md\vert}\mathcal{B}_{\Pi_{F,\infty}, \phi_{\infty}}\left(\frac{md^2}{c^3N^2}\right),$$
which looks quite nice.

Note that along those lines it is possible to derive many more explicit Voronoi type formulae for $\text{GL}_n$. However, in general the computational overhead might be far bigger.