You need to connect the payment system to the existing site

On the medical examination site, you need to add a step of accepting payment after the examination, but before the results of the examination are displayed.
The second option is that the visitor first pays for the examination, then proceeds to the already paid examination.

There should be an appointment form in which you can select one of the hospitals from the list, and follow the link to the website of this hospital.
You also need to develop a system / table that displays information about how many people have followed the links to the websites of hospitals and made an appointment.

All this functionality needs to be integrated into the appropriate places of the existing site and the users personal account on the site.