It's more than practice, Kharn. It's practicing with good form that gets hits. A coupla hints, from a guy that usually hits most of them, sometimes just more often than not.
First, pattern the thing. Every now and then even an 870 has a bbl that's bent a trifle, or sends its charge cockeyed for some reason. Once you've established that the shotgun is not the problem....
Get back to basics. While at home,and with an UNLOADED shotgun in a room that has no ammo in it, mount the thing and swing along the edge of the ceiling, or some other straight and more/less horizontal line. Move a bit and try a different angle. Get used to the "feel".
Now, on the range, keep proper stance and form. AND, the biggest reason I get into slumps is stopping the bbl when firing and not keeping the swing going. It's not just me, the reason trap guns are balanced more muzzle heavy than field guns is to add a little inertia to the swing.
Fine tune your focus. Instead of shooting at the whole target, try breaking the leading edge, or any other spot you can focus in on. When I can do this,and it takes more concentration than I can always summon, my score goes up. Same thing in the field, don't try to hit a dove, try to hit the dove's eye.
Hope this helps, and keep me posted....