It could just be heavy thick menses. When I was young, my first periods were like that -- where I passed thick soft clots. Since you missed having a period for a couple of months, then your uterine lining may have thickend up a lot. Once it began to shed during your period, you were seeing what amounts to a really heavy flow. The white line you refer to, is probably just a mucous string.
Regardless, talk to your doctor.
Regular ovarian cysts, no. Abnormal growths within the uterus, possibly.