  Let $\pi$ be a Hecke--Maass cusp form for $\rm SL_3(\mathbb{Z})$ with
normalized Hecke eigenvalues $\lambda_{\pi}(n,r)$. Let $f$ be a holomorphic or
Maass cusp form for $\rm SL_2(\mathbb{Z})$ with normalized Hecke eigenvalues
$\lambda_f(n)$. In this paper, we are concerned with obtaining nontrivial
estimates for the sum \begin{equation*} \sum_{r,n\geq
1}\lambda_{\pi}(n,r)\lambda_f(n)e\left(t\,\varphi(r^2n/N)\right)V\left(r^2n/N\right),
\end{equation*} where $e(x)=e^{2\pi ix}$, $V(x)\in
\mathcal{C}_c^{\infty}(0,\infty)$, $t\geq 1$ is a large parameter and
$\varphi(x)$ is some real-valued smooth function. As applications, we give an
improved subconvexity bound for $\rm GL_3\times \rm GL_2$ $L$-functions in the
$t$-aspect, and under the Ramanujan--Petersson conjecture we derive the
following bound for sums of $\rm GL_3\times \rm GL_2$ Fourier coefficients
\begin{equation*} \sum_{r^2n\leq x}\lambda_{\pi}(r,n)\lambda_f(n)\ll_{\pi, f,
\varepsilon} x^{5/7-1/364+\varepsilon} \end{equation*} for any $\varepsilon>0$,
which breaks for the first time the barrier $O(x^{5/7+\varepsilon})$ in a work
by Friedlander--Iwaniec.
