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This artilce should help: https://www.gardeningknowhow.com/ornamental/trees/tgen/repairing-tree-bark-damage.htm

I would treat the hole with a fungicide to help slow any rot. Also, if you can cover (do not plug) the hole to keep water out. Trees can live with holes in their trunks without problems as long as the bark and first green ring of wood (which is what keeps trees alive) is healthy. However, in this case, you say it has darkened; therefore, I would recommend obtaining adivce from an aborist that specializes in these types of tree issues.

The dying branches could be a separate issue from the split. If the wound does not show signs of decay or other problems, it is likely not the cause of the dying branches.

Fungus and pests are the most common reasons for branch die off in trees. Treat the tree with neem oil to kill any fungus or pests that may be on the tree.

If you do not see improvement in the tree after 2 weeks of spraying it with neem, you should have a professional come in and look at it. Tree diseases are hard to diagnose without actually observing the tree, so you would need someone who could come in person to take a look.

This article will help: https://www.gardeningknowhow.com/ornamental/trees/tgen/repairing-tree-bark-damage.htm

The following articles should be of some help to you: https://www.gardeningknowhow.com/ornamental/trees/tgen/what-does-a-dying-tree-look-like-signs-that-a-tree-is-dying.htm
https://www.gardeningknowhow.com/plant-problems/environmental/how-to-tell-if-a-plant-is-dead-and-how-to-recover-an-almost-dead-plant.htm

These articles may help you: https://www.gardeningknowhow.com/ornamental/trees/tgen/repairing-tree-bark-damage.htm
https://www.gardeningknowhow.com/ornamental/trees/tgen/what-does-a-dying-tree-look-like-signs-that-a-tree-is-dying.htm