My name was changed illegally, what do i do?

my mother and birth father don't really get on and on my birth certificate my name is their last names hyphinated. but i told all my life by my mom its just her name. when i recently tried to get my own copy of my birth certicate from my city hall i was rejected because i'm not using my legal hyphinated name, no properly legal ID. the illegal name is even on my social security card. no i call or go to seems to have the ability to help. someone help me before i strangle her for ruining my life