RJ:
If I will manually delete that node, the leader line will also be deleted
Aha! That certainly makes your problem more complex.
Dependency
Does the third-party app. create the leader line? From your screenshot, they don't appear to be part of a cell. I guess that the dependency engine may be invoked if the leader vanishes.
Can you post a small DGN example so we can analyse what you have?
Duplicate Text Nodes
Where are the duplicate nodes: are they stacked up on top of one another or elsewhere in the DGN model?
Regards, Jon Summers
LA Solutions