  Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maass
cusp form $\pi_1$ and $\lambda(m)$ be those of a $SL(2,\mathbb{Z})$ Hecke
holomorphic or Hecke-Mass cusp form $\pi_2$. Let $\mathcal{H}\subset[\![
-X^{1-\varepsilon},X^{1+\varepsilon}]\!]$ and
$\{a(h)\}_{h\in\mathcal{H}}\subset\mathbb{C}$ be a sequence. We show that if
$\mathcal{H}\subset \ell+[\![ 0,X^{1/2+\varepsilon}]\!] $ for some $\ell\geq0$,
\begin{align*}
  D_{a,\mathcal{H}}(X):=\frac{1}{|\mathcal{H}|}\sum_{h\in\mathcal{H}}a(h)\sum_{m=1}^\infty
A(1,m)\lambda(rm+h)V\left(\frac{m}{X}\right)\ll_{\pi_1,\pi_2,\varepsilon}
\frac{X^{1+\varepsilon}}{|\mathcal{H}|}\|a\|_2
  \end{align*}
  for any $\varepsilon>0$, and a similar bound when $|\mathcal{H}|$ is big.
This improves Sun's bound and generalizes it to an average with arbitrary
weights. Moreover, we demonstrate how one can recover the factorizable moduli
structure given by the Jutila's circle method via studying a shifted sum with
weighted average. This allows us to recover Munshi's bound on the shifted sum
with a fixed shift without using the Jutila's circle method.
