Requirement and design gaps and bugs on it

What does your Scrum Team say about this?  The Scrum Framework says nothing about bugs or user stories.  It just mentions Product Backlog Items.  How you chose to manage those items is entirely up to your team. 

Now I'll give you my personal opinion since it seems you were asking for them. 

Why do you need bugs and user stories?  Why not just have a product backlog full of the same item type? If you do not try to treat things as different objects, you never have this situation.  For example when a gap in requirements are discovered, just create something to describe the requirement and then treat in the same manner as everything else in the Product Backlog.  One simple step instead of a process of creating multiple items and then having to determine a way to manage them all. 

What should be the correct approach?

You've got work which apparently remains to be Done for the Product. It doesn't matter if you've only just discovered this now. If the Product Owner values it, he or she should order it as an item on the Product Backlog. That's it. Whether or not it is captured as a user story, defect, ticket, issue or something else is  immaterial.

Thanks Daniel and Ian for the clarification. I got the point.

Thank you!