My guess would be the ends are soaking up any moisture that sits on that coping.
You need to support them somehow before starting this.
Cut out to above the decayed wood.
Insert a repair section. It needs to be a very good match for the existing and you'll need to filler it to hide the join then re-paint the lot. Perhaps a bit of DPM between the repair section and the coping to slow the uptake of moisture.
Alternatively, you can make a feature out of it being different, use any old repair section at all [doesn't even have to be timber, so long as it can take the load] and clad over it with something like skirting board, mitred neatly at the corners, then re-paint the lot. Obviously skirting isn't intended for outdoor use so you will have to keep on top of the painting schedule if you don't want to be re-doing this too often.
They look as though the bottoms have been shaped to sit in a pocket hole to me.
You have 2 options. Repair or replace.
To be honest I would simply replace the full piece rather than try fitting a stub replacement. You will need to support the upper structure before removing and replacing each post in turn. Use a good quality hardwood thoroughly treated with preservative to help prevent this happening too quickly in the future. If it is indeed inserted into a pocket it mat be an idea to drill a small, angled downwards, hole on each side to help with drainage of any accumulated water in the pocket. Before pushing the post fully into the hole apply a bead of silicone sealant around the edge to try and form a seal to stop any rain water running in.
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