It is my understanding that the states that give out aid, must now follow the Federal Guidelines for Temporary Aid to Needy Families, aka Welfare. This is the case in my state. This means that there are an extremely few exceptions to the "standard" requirements for receiving any aid. Case workers do not have much flexibility. When applying for aid, there is a 45 day wait for TANF. If someone is working, even part-time, they are most likely not eligible for any TANF. If someone loses their job, depending on the reason, they may receive TANF in 45 days or they may be deemed ineligible due to contributing to the loss of their job. The economy is poor and there are not a lot of employment opportunities. So it can take someone weeks, if not months to find another job. The list for HUD "Section 8" housing is closed. For those already on the list, the wait is usually a minimum of 4 years long. Public Housing and other "affordable housing" options are a minimum wait of 2 years, on average. If someone cannot pay their rent and is evicted, there is a 4-6 week wait to get into "emergency shelter". There are religious based homeless shelters that can get people in quicker, but mothers with children cannot bring in boys over 10 years old. So if the mom has an 11 year old son, she will not be able to get shelter with her entire family. If a single mom with two children does receive TANF, it IS NOT enough to pay rent even at the affordable housing complexes, unless the rent is based on income (a lot of affordable housing complexes have a fixed rent amount that is lower than the rent for unsubsidized housing, but is not really all that affordable IMO). The state does provide childcare cost assistance if one is income eligible, but often there is a fairly large co-payment.
When someone (especially low-income) loses their job and income stops or is dramatically reduced, they can easily get behind on rent. A landlord does not have to accept anything less than the full amount. With a non-payment of rent, anyone can get a 72-hour notice. If someone doesn't leave their rental and challenges the 72-hour notice, they will lose in court and then they will have an FED (Forced Eviction) on their record, which makes it nearly impossible to find a landlord willing to rent to them in the future.
With all this said, from what I am reading, my state has a lot more resources than this mother's state. As someone said earlier, looking at what she did afterwards, I would guess she had mental health issues prior to the children's death. Without time to consider viable options combined with mental health issues, I can see how this tragedy could happen.