  Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for
certain weighted bilinear Kloosterman sums in polynomial rings over a finite
field. As an application, we build upon and extend some results of Sawin and
Shusterman (2022). These results include bounds for exponential sums weighted
by the M\"obius function and a level of distribution for irreducible
polynomials beyond 1/2, with arbitrary composite modulus. Additionally, we can
do better when averaging over the modulus, to give an analogue of the
Bombieri-Vinogradov Theorem with a level of distribution even further beyond
1/2.
