Sounds to me like the shooter is moving around and not staying in the same spot.
Also, if they were any good at shooting they wouldn't be doing the multiple shots business. They would be aiming at their target and popping one shot at it. The target is likely to move after the first shot, so the multiple shot sequences are a waste of bullets and time by someone who is frustrated and not much cop at what they are doing. They are hoping for a lucky shot with the second one onwards.
I wouldn't say it's time to move necessarily, but it must be frustrating if you have to get up early for work and someone is noisy like that at night. If you know what it is that they are trying to wipe out in the middle of the night you could lure it to your garden/yard and get rid of it or rehome it some distance away if that is not your cup of tea.