A user does not have the Add button under Training Material in a Training Requirement.
Create permissions are not granted for the Training Content Set object in the permission set of the user's security profile:
In addition to having Read, Create, and Edit permissions for the Training Requirement and Training ContentSet-Document objects, this is also needed for the Training Content Set object. This is because any time new materials are added, a new content set is created.
Support Portal Documentation: What Permission is Needed for an Admin User to Remove Training Material on a Training Requirement in Vault?