  In a pervious paper Weidmann shows that there a bound on the number of orbits
of edges in a tree on which a finitely generated group acts
$(k,C)$-acylindrically. In this paper we extend this result to actions which
are $k$-acylindrical except on a family of groups with "finite height". We also
give an example which gives a negative result to a conjecture of Weidmann from
the same paper and produce a sharp bound for groups acting $k$--acylindrically.
