David Burch has been the person to popularize this method of sight averaging...to the point where he is even referenced in Bowditch (2017), p. 304. I just have a couple of additional comments.
If you use any sight other than your first, your GHA will vary...and this in turn will cause your assumed longitude to vary. This is just fine, but you will want to ensure that your meridian ange doesn't change, so you don't have to rework your tablulated Hc and asimuth.
So then, let's suppose your GHA is 131° 03.0' and your assumed longitude is 166° 03.0' W. This gives you a meridian angle of 35° East.
You take a set of four sights, and you select the one that taken exactly 4 minutes after your first. The GHA is now 132°: 03.0'. You will want to move your assumed longitude to the west by one degree, to 167° 03.0'.
By doing this, you keep your meridian angle at 35°, and your Hc and azimuth are unchanged.
Of course, you could equally well KEEP you assumed longitude at 166° 03.0', and change your meridian angle
by one degree...but then you would have to redo your tabulated Hc and azimuth. This won't take much time, but
In the end, of course, it will all work out to the same fix for your location. How you get there will become a matter of personal preference. In my own case, I have gotten good results - in a situation where my first actual shot needs to be discarded - by using a hypothetical shot at the precise same time as my first actual shot.
The advantage of this is that all of the notations I have made on my worksheet remain valid. The only thing that needs changing is the hs value used in the center column at the bottom of the worksheet. There are no erasures or cross-outs that need to be done.