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Inspired by the work of Bourgain and Garaev (2013), we provide new bounds forcertain weighted bilinear Kloosterman sums in polynomial rings over a finitefield. As an application, we build upon and extend some results of Sawin andShusterman (2022). These results include bounds for exponential sums weightedby the M\"obius function and a level of distribution for irreduciblepolynomials beyond 1/2, with arbitrary composite modulus. Additionally, we cando better when averaging over the modulus, to give an analogue of theBombieri-Vinogradov Theorem with a level of distribution even further beyond1/2.

Bilinear Kloosterman sums in function fields and the distribution of irreducible polynomials