Bilinear sums with Kloosterman and Salie sums

We obtain new bounds on Type-I bilinear sums with Kloosterman and Salie sums and also with some similar sums. In particular, we improve some recent bounds on such bilinear sums due to V. Blomer, E. Fouvry, E. Kowalski, Ph. Michel and G. Milicevic (2014-2018). As a result, we improve the error term in an asymptotic formula for mixed moments of $L$-series associated with Hecke eigenforms. Finally, we outline several possible arithmetic applications of these bounds.