To "undermine" something means to make it weaker. You can "undermine" things like:

• someone's authority
• someone's argument in a debate
• the foundation of a building

When you "undermine" someone's authority, it means to do things that make people lose respect for them. For example, if you point out your boss's mistakes in front of other people, it undermines his or her authority.

In the example at top, the tech support manager wants all requests to go through him. He thinks that, if the requests go directly to his employees, it will make people respect him less. In other words, it will undermine his authority.