The eye is the lowest pressure area. When this bubble forms it shuts off all liquid coming into the pump suction. This could cause the pump to lose its prime. You cannot vent a running pump because centrifugal force will throw the liquid out the vent leaving the air trapped inside. Air is coming directly into the pump.
Negative suction happens when the pump is lifting liquid, pumping from a condenser hot well etc.
Air is coming into the stuffing box through the pump packing. This is a real problem if the pump has been fitted with a repeller to lower stuffing box pressure. Air is coming into the stuffing box through an unbalanced mechanical seal. As the carbon face wears the spring load holding the faces together diminishes. If you are using mechanical seals in vacuum service they should be of the O-ring design.