  The Cucker-Smale flocking model belongs to a wide class of kinetic models
that describe a collective motion of interacting particles that exhibit some
specific tendency e.g. to aggregate, flock or disperse. The paper examines the
kinetic Cucker-Smale equation with a singular communication weight. Given a
compactly supported measure as an initial datum we construct a global in time
weak measure-valued solution in the space $C_{weak}(0,\infty;\mathcal{M})$. The
solution is defined as a mean-field limit of the empirical distributions of
particles, which dynamics is governed by the Cucker-Smale particle system. The
studied communication weight is $\psi(s)=|s|^{-\alpha}$ with $\alpha \in
(0,\frac 12)$. This range of singularity admits sticking of
characteristics/trajectories. The second result concerns the weak--atomic
uniqueness property stating that a weak solution initiated by a finite sum of
atoms, i.e. Dirac deltas in the form $m_i \delta_{x_i} \otimes \delta_{v_i}$,
preserves its atomic structure. Hence they coincide with unique solutions to
the system of ODEs associated with the Cucker-Smale particle system.
