Mixed moments of twisted $L$-functions

We establish an asymptotic formula with a power-saving error term for the twisted mixed moment of Dirichlet $L$-functions and automorphic $L$-functions twisted by all primitive characters modulo $q$, valid for all admissible moduli. As a special case, this extends the asymptotic result of Blomer, Fouvry, Kowalski, Michel, and Milićević to general moduli, achieving an error term as sharp as the best bound recently proved by Khan and Zhang for prime moduli.