Hi @marilyn3
Over all you don't have a problem there. The treated posts should be H4 and are suited to occasional wet areas and will last many years.
Your concreter did a great job of sloping the water away from the house and has done a very neat job. Keep that person's phone no if you have further concreting jobs to do.
But I do understand you desire to free drain the puddles rather than let them sit.
Like you have already observed the posts with the open holes at the back drain faster. The holes or gaps formed naturally while the concrete was poured this is good as it lets water drain into the ground naturally as you already know.
So, if you look more closely at the posts with the puddles and the cement you will see the concrete is sitting higher with a little lip stopping the water from draining out the hole.
Solution
The simplest neatest solution is to chisel little bits out of the concrete around the timber posts so there is a little channel for the water to run down to the back hole and away. Especially where there is slight raise cement lips forming a dam.
To do this is easy A regular claw hammer and a low-cost Cold chisel designed for such application Use the Narrow chisel only.
The hammer and chisel will easily chip small 5mm to 10mm bit of concrete out touching the wood post. Chip around the wood post for a surface drain to the back.
So start at the back nearest the fence and make a little channel towards the puddle at the front. If you do this when you have water on the path with a hose you will soon see if the water is draining with your efforts use this as to your success guide..
Please wear safety glasses in case of Concrete bits jumping up.
Plan B is drilling holes through the concrete, but you will be left with unsightly holes. Holes can be drilled through the expandsion black strips. Or you can drill in between the concrete and the timber post.
For this you will need a long 300 mm cement drill bit 10mm and a hammer drill.
