Your best option is cutting back the siding so the corner board, whether you add a piece or remove and rip a piece of 5/4X6 to fit the new width.
If you choose to add a piece after cutting the siding, the gap looks like it may only need a piece of 3/4" material in the gap which will leave a small caulk gap to fill. Another suggestion I would like to make is, do not try to line the face of the added piece to match the board in place as in trying to make it look like one piece, it will not work. The caulk at that joint will smear over the face and change the texture of the corner board, just like the joint in the picture, where I would take a hand wire brush to, and texture that joint by pulling down only on the brush to remove some of the filler, leaving hopefully coarse lines to help blend the joint in. Since the joint is cut at an angle it will shed water, if the cut ends are primed, better still. Square cut butt joints will allow water to easily migrate in, but you do not have that here. So much for the side note.
The idea is to either recess the added piece to create a 1/4" reveal to the corner board there now, and still have the bevel siding a little behind the added piece still. That way the inside corners can be caulked easily without smearing all over the face of the trim.
If that won't work, do the opposite add a piece that is 1/4" thicker than the corner board to create a reveal over the existing corner board, giving the needed corner to caulk to.
Over all the best thing to do is to cut the siding and replace the corner boards ripped to meet the newly recut siding. Either way you go, bed the corner boards in caulk after any cut ends are primed. Not so much to make it ooze out all over, but enough to know the caulk is working "behind the scenes" and finish up the faces with caulk for painting.
Please realize, caulk can hold in water as well as keep it out, so be certain all caulk is filling what it needs to fill. Surface caulk will fail over time, the stuff you place behind your material is protected from UV and weather so it will outlast anything on the surface.