That is your alternate intake hose for winterizing (sucking in anti-freeze). You insert that into a bottle of RV antifreeze (IMPORTANT - USE ONLY non-toxic RV anti-freeze!!) Edit: Just noticed you are in AZ, so perhaps you never need to winterize...I'm jealous if that is the case...then again...I rarely have to use AC here...(cold, chilly Canada).
Notice how it connects to a valve (the circular shaped white piece directly beside the blue colored connector) - there is a little lever under there you can flip 90 degrees (when you need to winterize etc). It is currently in the "tank" position...which lets water flow from the fresh water tank into the pump. By flipping that valve (pointing towards the short hose) the pump will now pull from the open end and the pump no longer draws from the water tank.
If you find it is running often, you may have a leak somewhere. Turn on the pump and just check everywhere possible - hot water tank area, toilet, shower (and the valves behind the shower controls), under the sink etc. Any slow leak (a dripping faucet or shower head, or a little dribble of water into the toilet etc.) will cause the pressure to drop and then eventually the pump ramps up the pressure again.
That said, I had the same problem and I could find nowhere it was leaking. I eventually removed the pump (thinking it was faulty) and completely disassembled it (it is pretty easy....just carefully note where all the wires and screws go) and cleaned/checked the rubber diaphragm inside (I found nothing obviously wrong) but after re-assembling it all the problem went away... I assume there was maybe some debris or something not sealed correctly.
Hard to tell from the picture, but you might even be able to disassemble the pump right there without disconnecting any hoses or removing the pump at all. All the screws seem to be accessible. Someone more knowledgeable here may have some advice.
Here is a video that goes over some disassembly....
https://youtu.be/XaJjU09ygmU