You don't keep them. The tables and relationships represent your business rules/facts.why do you keep the many to many relationship between dogs and handlers and the many to many between handlers and agencies
You indicate a Handler can handle several Dogs and a Dog can be handled by many Handlers.
Similarly, a Handler can be associated with many Agencies, and an Agency can represent several Handlers.
In database diagramming you break the many to Many relationships into multiple 1 to Many relationships.
If the tables and relationships in the draft model don't match or meet your business requirements/facts, then adjust as necessary. There is nothing sacred/permanent with the draft model. It was intended as a starting point for consideration based on my understanding of your post.