This injecion is called Depoprovera which is commonly used for contraception. It is a type of progestogen, which you thought would increase the risk of cyst rupture. After injection, you won't see cyclical menstruation but persistent spotting instead. You are right, it could be annoying but, with time, you won't have menstruation at all. Weight gain is not a common problem. Occasionally, women might feel depressed.
You can go and read up about how this endometriotic is being formed, you will then understand why the chance of rupture after taking the progestogen is not high. Progestogen would cause thinning of endometrial tissue. The inner lining of the cyst is endometrial tissue, but, the outer layer is normal ovarian tissue. Therefore, thinning of inner layer of the cyst theoretically might increase the risk of rupture but really uncommon. If you use Depoprovera after operation, it may reduce the chance of recurrence but there is no guarantee. Using the drug now doesn't help because the cyst is too big.