Degradation in performance of pump
Are you talking about a pump like this in the photo? It's rated (advertise in AliExpress) to use 130-220 mA, which is too much for a 9V battery. At 9v the pump will draw even more current than this. 

A 9v battery can supply about 450 mAh, but it will be more comfortable with a lower current.

This behaviour that you are describing (~ 5 min untill the performance degrades) seems to be normal, when using this battery. Probably the 7.8V reading you talk about was taken with the pump off, but with the pump on the voltage could be even lower.

There's no fault to the pump or the battery, maybe it's just because they're not compatible. I suggest you to try another power source for your project. A wallwart from a old cell phone could be perfect. If you really need to use batteries, try using a batch of three or four AA cells.

By the way, those pumps are rated with a very low lifespam, just about 500 hours.