It depends how extensive and the severe the tear is. The supraspinatus muscle is a part of the rotator cuff, the impairment of which could arm dysmobility. In fact, supraspinatus muscle tear is the most common part that is injured in rotator cuff injury.
Supraspinatus muscle tear cases could be divided into partial thickness tear and full thickness tear.
Depending on the severity of your tear, your surgeon may recommend starting with a non-surgical treatment like physiotherapy and supraspinatus tear exercises with the goal of restoring the range of movement and the strength of the shoulder, activity modification (changing your lifestyle to avoid the tasks that increase your pain and may exacerbate the tear), pain management with non-steroidal anti-inflammatory medications, analgesics, and cortisone injections.
Of course, all these options should involve regular check-ups with your orthopedic surgeon in order to make sure the problem isn't getting worse.
For full thickness tears and more major tears (or if the tear involves more than one tendon) or there is significant damage to the tendon, various surgical procedures may be required.