The power of influence occurs in personal, positional, and relational power bases when strategies for influencing events and outcomes are employed. In the power of influence, it is important to assess when particular influence strategies are failing and to adapt to a different strategy of influence.
Empowerment through delegation is a different sort of thing in that while the power of influence is a direct effort on the leader's part aimed toward a particular party, empowerment through delegation is a deliberate effort on the leader's part to remove themselves from the line of influence and empower another selected individual to exert the influence from their own unique base of power.
The power of influence is a direct exertion of influence by the leader. Empowerment through delegation is the deliberate inroduction by the leader of a different source of influence.