  We bound non-linear additive twists of $\mathrm{GL}_{3}$ Hecke eigenvalues,
improving upon the work of Kumar-Mallesham-Singh (2022). The proof employs the
DFI circle method with standard manipulations (Voronoi, Cauchy-Schwarz,
lengthening, and additive reciprocity). The main novelty includes the conductor
lowering mechanism, albeit sacrificing some savings to remove an analytic
oscillation, followed by the iteration ad infinitum of Cauchy-Schwarz and
Poisson. The resulting character sums are estimated via the work of
Adolphson-Sperber (1993). As an application, we prove nontrivial bounds for the
first moment of $\mathrm{GL}_{3}$ Hardy's function, which corresponds to the
cubic moment of Hardy's function studied by Ivi\'{c} (2012).
