I corrected a similar problem on a friends bow the other day. I shot with him at a tournament and he couldn't hit the side of a barn with his bow. The bow was his first and only attempt at making a selfbow.
When I saw how poorly his arrows flew I asked to look his bow over. He is right handed and the string was at least an inch off bows handle to the right side.
I explained how he could fix the problem but he asked if I would take the bow home with me because I had more experience with a heat gun than he did.
I tweaked the tip on the upper limb to the left first as it has a slight dogleg to the right. This barely got the string on the handle. Next I used my contour gage to check for asymmetry side to side on the top in the fade out area, one side was much thicker than the other so I evened things up. Getting all the wood symmetrical will often get the string down the middle.
Here is my contour gage, invaluable for making sure the belly on your bow is even side to side. They cost about $7 at Lowe's.
My next step was heating the handle at the arrow pass and moving the top limb to the left just a little. This did the trick and the string was just right over the handle and the bow started launching arrows like it was supposed to, right down the middle.