  Let $\lambda_g (n)$ be the Fourier coefficients of a holomorphic cusp modular
form $g$ for $\mathrm{SL}_2 (\mathbb{Z})$. The aim of this article is to get
non-trivial bound on non-linearly additively twisted sums of the Fourier
coefficients $\lambda_g (n)$. Precisely, we prove for any $3/4 < \beta < 3/2$,
$\beta \neq 1 $, the following non-trivial estimate $$ \sum_{n \leq
N}\lambda_g(n)\,e(\alpha\, n^{\beta})\ll_{g, \alpha, \beta, \varepsilon}
N^{\frac{1}{2}+ \frac{\beta}{3} +\varepsilon} + N^{\frac{3}{2}-\frac
{2\beta}{3} + \varepsilon}, $$ for any $\varepsilon > 0$. This is the first
time that non-trivial estimate for such sums is achieved for $1 < \beta < 3/2$,
breaking the barrier $\beta = 1$ in the work of X. Ren and Y. Ye. It also
improves their estimate in the range $9/10 < \beta < 1$. The key of our
approach is a newly developed Bessel $\delta$-method.
