What damage general anesthesia in the mom do to an unborn child?

None, most likely. A number of anesthesia medications have a long record of safe use during pregnancy, with no evidence of any kind of harm to the baby. We try not to give general anesthesia during the first 3 months unless it is an emergency. Be sure to tell the anesthesiologist you are pregnant. Depending on the type of surgery, spinal or epidural anesthesia may be a very safe choice.
Decrease blood flow. Th main concern in anesthesia is a decrease in blood pressure in the mother that can lead to a decrease of blood flow in the placenta. This can be easily treated and managed by your anesthesiologist. There is also a concern that anesthesia and surgery might induce early labor. Again this can be monitored and treated. Teratogenesis is possible in early pregnancy. I hope all goes well.
It is a risk. People never undergo anesthesia without subsequently undergoing a surgical procedure. So the effects of anesthesia on a pregnant woman and her unborn child are actually the combined effects of anesthesia together with surgery. •the chance of miscarriage or fetal death as a result of anesthesia and surgery is about 5.8% when averaged over all trimesters. It is 10.5% in the first trimester.