Does Bellman–Ford algorithm can handle zero-weight cycles? According to my understanding, the Bellman–Ford algorithm can handle cyclic graphs with negative weights. but it cannot have negative cycles. But can it handle zero weight cycles?
Yes.  Bellman-Ford can handle graphs with zero-weight cycles; they aren't a problem.

Intuitively, negative-weight cycles are problematic because they can make the notion of "shortest path" ill-defined: there is no shortest path.  For instance, suppose we have a cycle from A to A with weight -5, and an edge from A to B with weight 10.  Then there is no shortest path from A to B.  There's a path with length 10 (go straight there), a path with length 5 (traverse the cycle once, then go there), a path with length 0 (traverse the cycle twice, then go there), a path with length -5, a path with length -10, and so on.

Zero-weight cycles don't cause this problem.