This is not a good situation for the pump. I assume you are talking about a centrifugal pump. Without more information about your system, it is difficult to make an assessment. But a few thoughts come to mind.
Suction Problems - If the pump is trying to achieve suction lift from a sump below the pump elevation, it could have an air leak that results in an air bubble that has to be displaced before full performance is achieved. This could be very damaging to the pump. Bushings and wear rings in the pump are designed to be full of liquid. If they are passing vapor while the pump regains prime, there could be excessive shaft deflection, rubs, high thrust loads, etc.
Discharge Problems - If the discharge line is not liquid packed at start-up, there will be a delay while the pump floods the line up the flow meter. A typical flow meter in our plant is an orifice plate with a DP measurement that correlates to flow. With gas passing through the orifice plate, there won't be enough pressure drop to register any flow. In this case, the pump is actually pumping at a very high flow rate until the line fills. This can also damage the pump. Running at higher flow brings along a higher Net Positive Suction Head required. The pump can cavitate during this period. There can also be higher radial and thrust loads depending on the configuration of the pump. In a vertical turbine pump, the shaft could buckle. In a horizontal multistage pump with a balance piston for thrust compensation, the thrust bearing could be overloaded and fail.
If you could describe the configuration of the pump and the system you would get more useful replies.
Johnny Pellin