I'm in the US and we have a different aid system and different social stratification, however, my mother was a single mother on welfare after my parents divorced, which is clearly different from the situation you are suggesting, but here is a problem she encountered as far as working goes.
As she had minimal upper education, and had spent the better part of her adult life raising me and my siblings, she did not have much earning power in the work force. She could make $10 an hour tops, at the time, if she worked full time.
That's about $1341.6 dollars per month after Federal, and not state taxes.
Our rent for our tiny house (my brother had to sleep on the couch) was about $1500 a month.
Food was about $900 per month, not including for me because I bought my own.
Electricity was $60 per month.
Gas was $30 per month.
Phone was $60 per month
Gasoline would have cost her around $90 per month at the time.
And car insurance about $30 per month.
This amounts to $2511.60 and does not include health insurance/health care, household items, personal care items, school supplies, or clothing.
To send my brother to daycare would have cost around $258 a month.
She would have had to bring in about $3,000 a month after taxes to meet basic needs.
She did not get $3,000 a month in welfare. She got less than $1500 and a relative helped out on behalf of my father, who could not pay child support at the time. We were still living below the poverty line and often did not have at least one utility at any given time....I used to have to ride my bike to the gas station, where there was a payphone, to make phone calls, and then they tore the gas station down, and the phone with it.
She tried to get a job but could only find one that was part time. The amount she would have paid in gas and daycare would have eaten most of her earning up. She decided it wasn't worth being an absent parent to only bring home an additional $40 a month or so.
Remember, raising a child is a full time job. They don't raise themselves so it's important at least one parent be present most of the time.