  We consider special flows over the rotation by an irrational $\alpha$ under
the roof functions of bounded variation without continuous, singular part in
the Lebesgue decomposition and the sum of jumps $\neq 0$. We show that all such
flows are weakly mixing. Under the additional assumption that $\alpha$ has
bounded partial quotients, we study weak Ratner's property. We establish this
property whenever an additional condition (stable under sufficiently small
perturbations) on the set of jumps is satisfied. While it is classical that the
flows under consideration are not mixing, one more condition on the set of
jumps turns out to be sufficient to obtain the absence of partial rigidity,
hence mild mixing of such flows.
