  Additive twists are important invariants associated to holomorphic cusp
forms; they encode the Eichler--Shimura isomorphism and contain information
about automorphic $L$-functions. In this paper we prove that central values of
additive twists of the $L$-function associated to a holomorphic cusp form $f$
of even weight $k$ are asymptotically normally distributed. This generalizes
(to $k\geq 4$) a recent breakthrough of Petridis and Risager concerning the
arithmetic distribution of modular symbols. Furthermore we give as an
application an asymptotic formula for the averages of certain 'wide' families
of automorphic $L$-functions, consisting of central values of the form
$L(f\otimes \chi,1/2)$ with $\chi$ a Dirichlet character.
