Hello, I can assist you. Without doing an on-site structural review it sounds like you can tuck point that gap the whole way up. The stucco itself is not structurally sound enough in my opinion. I recommend you do both. I'm not sure of your masonry experience level but it is not a difficult repair.

First you need to remove the old material from the joint, this will be time consuming but it has to be done. You can grind it out but I recommend just taking a chisel and hammer and chiseling it away. If it is a very old repair it won't take a terrible amount of time.

Once the joint is free of debris you can take your mortar mix and whip it up to a consistency a little thicker than cake batter, fluid enough to move around but thick enough to where you you can score a line in it and it stays up.

Take this mortar and put a little on a block and put the block up against the brick ledge, take a tuck point trowel (which is like a regular trowel but long and skinny) and force the mortar into the join until it is flush with the surface, you want to get at lease an inch of it in there. Wipe away any mortar that is left on the brick with a wet towel.

joint*

Do this the whole way up the wall.

I recommend you use regular off the shelf mortar mix from Lowes or Home Depot. If you use a stronger mix you run the risk of cracking the brick. You always want to use a mortar that is weaker than the brick.

Do not allow the mortar joint to get too dry for about three days, it takes time to properly cool. If this is on a south facing wall I recommend misting it a few times a day for three days. If it is a north facing wall you can only mist once a day. Just keep it damp. If it dries out before it cools it will be prone to cracking and you will be back where you started.

Once it has cured you may stucco over the wall as normal. I hope this helps, let me know if you have any more questions.

cures* not cools, gotta love instant chat