Within the Weekly Review in the Payroll tab, users can edit an employee's payclass as long as they have the appropriate permission. 

Within the Security Roles section, a permission 'Edit Employee Payclass in Payroll Tab' exists. Toggle this permission to ON and the user can now modify an employee payclass. 

Note this permission can be set per project so it is possible for a user to be able to change payclasses for employees that have worked on one project but not another project. In the image below, the user has access to modify payclasses on project 20-003 but not on project 19-001. Thus, on 19-001, clicking on the payclass field does not result in a pop up window where the payclass can be modified.