Although the Supraspinatus is an integral part of therotator cuff in its role in initiating movement, you can function with a full tear. However most commonly I find a lot of people then suffer secondary symptoms where the surrounding muscles start to give a little too much of themselves and trigger points develop and effective shortening of the muscles occur. The Upper Trapezius fibres and Pectorals tend to be prone to this. After some time pain and shoulder complex dysfunction come into play.
If I was your Physio I'd start by dealing with any trigger points in all the prime suspects (and there are a few!). There are several effective techniques, such as acupunture, TP release, myofascial release. Posture correction is appropriate as is ensuring good Scapula control. Then you can start to look at (cautious) strengthening. If you're into doing weights I'd also be careful in not overloading the muscles you've had released. Loss of form and technique, and early fatigue in the shoulder are good pointers.
Ultimately it could be a stop gap and surgery might be the only answer. Have a look shirtless in the mirror. Does your shoulder roll forward or does it look like the 'ball' of the joint has dropped? They both [I]could[I] be an indication of bad the situation is.
All the best with your future treatment,
Sweens