Within NaturalHR it is possible to allow managers to add training on behalf of an employee. To do so navigate into:

Administration -> Company -> Settings -> Display Settings

Once on this page just scroll down to the Manager Views Section -> Manager Module Permissions and then click the Add training, edit training and delete training checkboxes to select all desired permissions.

Once they're selected click Submit.

Once this is submitted managers will now have permissions to add new training and edit or delete existing training records: