Yes a PSP is an initial push then converts to a policy. Now depending on how you configure the policy will dictate what happens on the policy level. Is your delivery method set for Required - Once? Also, if there are any machines still being processed from the push I don't think it will convert into a policy - hung processes on machines, etc...
Yes, the delivery method is configured for required-once. What happens is I start the job, all the machines to into "active", within a minute or two (after the status of the machines has been determined) all the offline machines go into failed. These are actually offline (or currently off-site) machines, not hung.
In 8.8 they will go into a failed state if they are off during the initial push. Once they fail the policy gets marked as active and you have to wait for the clients to check back to the core server to see if they have any policies to run.
If it is taking a few days for the clients to get the policy then it is probably 1 of the following:
1. the device has been offline for a few days.
2. the agent was not on long enough to run its policy sync. For example you have it set to run policy sync every 24 hours. It maybe trying to run the policy check at 8:00 PM but you are shutting down the pc at 5:00pm. So the client was not on when it was scheduled to check for new policies.
You can always check the pc to verify that the Policy is able to get to the pc by manually launching the Software Distribution Portal on the client. This will force the pc to go out and look for new policies.
You might want to consider putting some time restraints around when you want the computer to look for new policies. such as 1 every 8 hours. Or perhaps to only kick off once a week during your maintenance window.