The best thing that can help accuracy is having a good gunsmith "back bore" it and "lengthen or extend the forcing cone".
The forcing cone is the portion just in front of the chamber where it necks down slightly from a larger diameter where the shell fits to the true bore diameter.
As a stock shotgun intended to scatter shot (not what you want for Turkey hunting) on most shotguns the length of the forcing cone is short so it funnels the shot down to a tighter diameter very abruptly and quickly. The end result is the shot load is actually bouncing around inside the barrel as it travels the length of the barrel and when it leaves the muzzle this bouncing also causes the shot to spread out quicker.
It's a case of forcing too much shot into too tight/small of an opening all at one time and is actually counter productive to what the Turkey hunter is trying to accomplish.This also CAN cause some of the shot to loose it's shape from quickly being forced against the other shot and barrel, more common when using heavy shots loads as far as oz. of shot.
When you lengthen or extend this forcing cone the shot load has more room to stretch out as it pass through the barrel. The result is less bouncing around, banging against each other and the shot travels down the length of the barrel in a much more controlled straighter path thus reducing how much and how quickly it begins to spread when it leaves the muzzle resulting in a tighter pattern.
Years ago I took my 870 Remington to Mark Bansner in Adamstown Pa. as Mark is world known for his custom built Turkey and slug guns as well as custom rifles.
Mark took a great deal of time to actually show me this and explain it to me and showed results of before and after from many guns and the results were amazing.
As he explained this is why in many cases a shotgun will actually pattern better with a lighter weight shot charge such 1 1/2 oz. of 4 shot than it will with a heavier load such as 2 oz. of 4 shot. Even though the 2 oz. load has more shot because your trying to shove more shot down the barrel with the 2 oz. load this causes it to be funneled down to quickly in the forcing cone and results in deformed shot and a pattern that opens up quicker.
Not a concern with most shotguns as they are used for shooting moving targets but it is a concern for us Turkey hunters.
He also explained how the best pattern is achieved by a balance of the proper length of the forcing cone and the proper size choke diameter. He said how he hears of so many people who buy the tightest screw in tube they can find only to discover that their gun patterns worse than it did with a larger diameter tube.
Again he explained how it's the principal of trying force to much shot through to small of an opening at one time and in reality in these cases a lighter shot charge will pattern better but the human mind is geared to think the opposite.
Once he reworked my 870's barrel the pattern tightened tremendously from using it stock with a super full Turkey choke in it.
He back bores it, lengthens the forcing cone,cuts the barrel shorter to around 20" and machines the muzzle to a permanent choke diameter.
He explained that maximum velocity from a shotgun is achieved by a 20" barrel and any longer is actually slowing the shot down slightly. The longer barrel does provide a greater sighting plane for a wing shooter but since using one for Turkey hunting is typically shooting it more so as a rifle at a standing target and the Turkey hunter typically slings his gun over his shoulder and encounters brush he cuts the barrel short as close to 20" as possible.
I know he took my 870 special purpose Remington that already patterned what I thought was good out of the box and made it an awesome Turkey killing shotgun.
The only down side I've discovered is that the pattern is so tight that if the bird is in close say within 25 yds or so you better be dead spot on or your likely to miss.....just ask my Dad & Austin my Son....
Hope this explanation makes a little sense, was kind of hard for me to put the technical information into words.
Here's a link to Mark's web site.
www.bansnersrifle.com/index.html