Welcome to GBO! Check to see if the latch has oil on it, one of the most common problems that H&R's gunsmiths see is oil on the latch which may cause inconsistent accuracy. Most of us store our rifles muzzle up, that allows the oil in and on the barrel to run down into the frame and on the latch and latch shelf. Make sure they're dry and oil free before you shoot. Some have reported that putting oil on the latch helped to improve accuracy, so that's always an option, but H&R recommends it to be dry.
Have you shot the rifle without the forend on? How's the barrel break open? I like mine to have just a little resistance when opening, so the barrel drops open by itself, but not loosely. If it's real tight, remove the forend spacer and flat sand the surface the fits against the forend, this will reduce the pressure the forend puts on the barrel and frame. If it's too loose, place a shim tween the spacer and forend, I use aluminum furnace tape, it's not expensive and it's impervious to weather, I cut it just a tad smaller than the spacer so it can't be seen.
Other options are to bed the forend, one easy method is with RTV silicone which is covered in the FAQ, but permanent bedding will work too.
Other possible issues are bad scope, ring or base problems, you'll know more once you've check the issues I've pointed out, don't assume the scope is not to blame, it happens.

A trigger job is also in your future if you want the best accuracy, not hard to do following the instructions in the FAQ, many novices here have done them with excellent results.
Hope this helps,
Tim