Need advice. I have little patience for my daughter.?

I am a single mother who is studying full time and working part time. My daughter is 2yrs and 4months old. I desperately wanted a child for about 9yrs before I got pregnant with her, the pregnancy was hard emotionally and I became suicidal. There was complications during labour and when she was born she was taken from me and I didnt get to hold her for a day. After that it was only her father who could really settle her. I left her father when she was 1.5yrs old. She goes to him every fortnight and they have a very close relationship. This year I have considered letting her go to live with him as I think she would be happier. I dont have a social life, all I do is work, study and be with her. My mother has her for a few hours every fortnight or so. Other than that we dont see anyone or have other friends with kids. She is very demanding and constantly asks for something even after Ive said no and I find this is when I end up losing my cool. She talks a lot, repeating everything until Ive acknowledged it, cries a lot and has a lot of tantrums. I have started to scream at her and lately I want to hurt her like today when she was having a tantrum at the shops I dug my fingernail into her wrist while I was picking her up. I know that is child abuse and I do not want to feel this way, I want to be able to enjoy our time together but I feel she makes it so hard. What do I do?