White discharge with heavy itching and a scratched, sore-feeling area usually points to a vaginal infection, most often a yeast or bacterial one, made worse by scratching. The egg allergy you've listed isn't related to this.
Next Steps
Avoid scented soaps or wipes down there, wear loose cotton underwear, and keep the area dry. A gynaecologist can examine you and send a swab test, which identifies the exact infection so the right antifungal or antibiotic cream gets prescribed.
Health Tips
See a doctor sooner if you notice a foul smell, bleeding, fever, or the skin starts to blister or peel.