I have several rifles that have bores that look like they are pitted to some extent. I cleaned them good, and with good ammo they shoot quite well.
I would advise before doing all the other stuff slug the bore first. The cause of inaccuacy in many of the Mosin-Nagant rifles is an oversized bore. I have one like that. It shoots 5 inch groups at 100 yards with the spam can ammo. With my hand loads using an oversized bullet it cut those groups to 2 to 3 inches. (My vision with iron sights limits this. Someone better beihind the trigger with decent eye sight could shrink that even more.
Now that said for a good cleaning that will help. Get some J&B Bore Brite. Use a cleaning brush. Oil the bore. Dampen a patch with oil. Rub it across the surface of the bore brite. Wrap the patch on the brush. It should be real tight fit. Go in back and forth in and out short strokes all the way down the bore, and back out. Rinse the excess with hot tap water. Run dry patches through till it is dry. Oil it again. See how it looks. It may take 2 or three times to get it better.
Note that if you slug it, and it is .310 get some brass cased ammo. Fire it out of the rifle. Buy a Lee Loader. (The $20 whack a mole.) and non metal mallet. Add a block of wood. Now buy some .311 sized bullets that are for the .303 British, and some powder, with some large rifle primers. Follow the directions, and use a powder that the instructions list that will work with the dipper. Your groups will shrink.
No matter how many times you do it and nothing happens it only takes something going wrong one time to kill you.