I would say the verge needs removing, cleaning and rebedding and pointing. If you just try to point that up it will drop out again.
You could consider a dry verge system.
It's not ideal but I don't have a big issue as a stop-gap measure. But it is important to get the right mix - sharp sand and dry as you can manage (as you would bed a verge).
A weak builders sand mix may work in a wall pointing, but not in a verge.
I’ve done these as a 2 coat application before, a bit like a render scratch coat first, pushed in well back from the edge, scratch keyed, then when dry, can do the second coat and really compact it in as first coat stops it moving back. Then give it a really hard iron with the trowel as it goes off. I know I’ll get laughed at by some of you but it does work, and saw it done previously by some old boys! Same mix as Jeds mentioned above.
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