It doesn't sound like your messing your child up at all. All children have tempers, sometimes bite their nails and so on. I remember once feeling like you describe, everytime my child felt sad or worried I was convinced I had passed on anxiety disorder and messed them up due to my own anxiety. Can you get some cognative behaviour therapy to help you recognise these negative thoughts? Try joining the mood gym online, it's free and a really good starting point.
All mothers worry about how they effect their children, that's love and instinct. Getting a bit of perspective would really help you be kinder to yourself and feel better in the long run. Its like a vicious circle. For me I had anxiety and depression, which I believed made me a terrible mother, which then depressed me further so I had no motivation for anything and couldn't see past my own thinking. I now look back at the time I felt like that and I can see I was sort of mentally bullying myself. I never credited myself for the good things I did, I simply obsessed about how I was failing. It go so bad I worried if I really was cut out for being a mum. These days couldn't be more different. I feel confident as a parent, I can see how my nurturing ways really help my children thrive.
I know it's difficult and really can feel hopeless. But you can change your perspective over time. I know how patronising it is when people say think more positively. For me I couldn't force the negative thoughts out, that made them worse. I had to let them in but correct them.
Can you start by giving yourself credit where credit is due? Each time you accomplish something, recognise it. Each time you get them to school on time, each time you shower or bath them, washing clothes, hair cuts, talking with them when they are upset, homework, Christmas fun and presents, cooking a nice meal, buying them a treat, when your kind etc. Praise yourself.
Now the next step is just as important, if not more so important when your depressed. You have to learn to be more kind, less judgemental and forgiving to yourself. Your a mother yes, but you will make plenty of mistakes. At times you may struggle, that is ok. So If you only give your child beans on toast 1 night, or forget to do homework, forget to wash clean shirts for school, don't feel like going out or happy and joyful then that is ok. There really is so much pressure at times, we see on Facebook how other mothers are and it's hard not to question ourselves. But what we don't see is the many tears that mother may have shed, her down days and mistakes. It's rare a mother stands up and says today I left the house a mess, had no motivation and my kids got on my last nerve and I may of shouted at them and now I feel bad.
Society presents to us the perfect mother, the ideal. Well I can tell you she doesn't exist! A happy mother still struggles, but she has learnt to accept that herself anx not beat herself up over things.
Oooh I do hope I haven't made u feel patronised, it's difficult to articulate but I just wanted you to know you can feel better and you can move forward. 