  We evaluate some twisted fourth moment of Dirichlet $L$-functions at the
central point s=1/2 and for prime moduli q. The principal tool is a careful
analysis of a shifted convolution problem involving the divisor function using
spectral theory of automorphic forms and bounds for bilinear forms in
Kloosterman sums. Having in mind simultaneous non vanishing results, we apply
the Theorem to establish an asymptotic formula of a mollified fourth moment for
this family of $L$-functions
