On your question about deleting relationships or nodes, here are examples of how you can do this using your example.
Deleting a node (Note: DETACH DELETE is required if there are relationships with the node(s))
MATCH (n:Name {name: "xxx"})-[r:HAS_STATEMENT]->(s:Statement {text: "blah blah"})
DETACH DELETE n
Deleting a relationship
MATCH (n:Name {name: "xxx"})-[r:HAS_STATEMENT]->(s:Statement {text: "blah blah"})
DELETE r