  In this paper, we use the Bessel $\delta$-method, along with new variants of
the van der Corput method in two dimensions, to prove non-trivial bounds for
$\mathrm{GL}(2)$ exponential sums beyond the Weyl barrier. More explicitly, for
sums of $\mathrm{GL}(2)$ Fourier coefficients twisted by $e(f(n))$, with length
$N$ and phase $f(n)=N^{\beta} \log n / 2\pi$ or $a n^{\beta}$, non-trivial
bounds are established for $ \beta < 1.63651... $, which is beyond the Weyl
barrier at $\beta = 3/2$.
