  In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed
graphs and digraphs. For digraphs, they showed that for every $\alpha>0$, there
exists a constant $C$ such that for every $n$-vertex digraph of minimum
semi-degree at least $\alpha n$, if one adds $Cn$ random edges then
asymptotically almost surely the resulting digraph contains a consistently
oriented Hamilton cycle. We generalize their result, showing that the
hypothesis of this theorem actually asymptotically almost surely ensures the
existence of every orientation of a cycle of every possible length,
simultaneously. Moreover, we prove that we can relax the minimum semi-degree
condition to a minimum total degree condition when considering orientations of
a cycle that do not contain a large number of vertices of indegree $1$. Our
proofs make use of a variant of an absorbing method of Montgomery.
