Moment estimates for the exponential sum with higher divisor functions

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k n} 1$. This follows from a quick application of circle method. Along way, we find minor arc bounds exponential sum with $\tau_k$, and asymptotics high moments Dirichlet kernel.