There are many people like you.
The good thing is you WANT to work, so many actually choose the life you are living and prefer to exist on whatever hand outs they can get or rely on family etc....
Keep putting yourself out there, jobcentre, internet, handing in your CV on the high street. A seasonal position may come up this time of year, get your foot in the door and see how it goes.
I would imagine that with the little one you will be pushed to find employment that suits childcare etc it is a difficult balancing act. My partner feels how you do sometimes as she can only bring in around15% of our household income due to having a child and feels that she burdens me. But hopefully, like us your relationship is strong enough to get through life's tests. My partner assists in other ways. Our home is always clean, washing done and food on the table. She makes our house a family home as well as working 3 days a week and that is our balance of things and it works for us.
I dont resent the situation one bit because I love my family and our home, I am happy to work 50hrs a week and bring in 85% of the income.
Just hang in there, like alba says, life throws these curve balls at you.
But it is important that you staf focussed.
Why would a minimum wage job not work for you, out of interest? If you want to get out the house and earn some money? It is a starting point is it not and lessens the gap in employement on your CV.