Is it permissible for an injured employee, who has been granted leave with continued pay to recover, to work during the leave period while still under the company's responsibility?

There is no impediment to working another job during a leave, because the employer only owns the employee's benefit during official working hours. Outside of these hours, the employee has the right to dispose of their time as they wish. A private employee (staff member) may take sick leave if they are entitled to it according to the contract.