A note on exponential-Möbius sums over $\mathbb{F}_q[t]$

In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, that $\max_{ θ\in [0,1) }\left|\sum_{ n \leq x } μ(n) e(n θ) \right| \ll_εx^{3/4 + ε}$. The purpose of this note is to deduce an analogous bound in the context of polynomials over a finite field using Weil's Riemann Hypothesis for curves over a finite field. Our approach is based on the work of Hayes who studied exponential sums over irreducible polynomials.