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Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maasscusp form $\pi_1$ and $\lambda(m)$ be those of a $SL(2,\mathbb{Z})$ Heckeholomorphic 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}^\inftyA(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 arbitraryweights. Moreover, we demonstrate how one can recover the factorizable modulistructure given by the Jutila's circle method via studying a shifted sum withweighted average. This allows us to recover Munshi's bound on the shifted sumwith a fixed shift without using the Jutila's circle method.

Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets