There's absolutely nothing wrong in offering advice to the OP. The more suggestions she gets, the more likelihood there is that something might just help her. But at the same time, those offering advice have to bear in mind that everyone's circumstances are different and not everyone will be able to avail themselves of the advice offered - as much as they'd otherwise like to. Not for nothing is the "poverty trap" called the poverty trap and sadly, disgustingly (for a so-called civilised society) there are many many people whose desire to "better" themselves so they can support their families with a basic standard of living means nothing if they literally can't take the steps necessary to improve their lot.
I think it's relatively easy for many people who've never been in desperate dire straits to think that improving your life is always possible one way or another, even if it involves sacrifice - but fail to recognise that for some families there is no way out. "Moving" - to areas with better employment prospects, and "retraining" - in the hope better qualifications and/or skills will lead to a better job are, not surprisingly, always touted ..... but I can't help thinking that some people have absolutely no concept of how impossible it is for some families to realise this. Moving: how do you transport your worldly goods from A to B ? How do you do this if you have absolutely NO savings and have already pared expenditure to the bone so you can't cut back any further ? Not everyone knows "someone with a van", not everyone has a credit card with which to pay - or, alternatively, cannot take on any further debt as they'd be unable to service this. How do you pay your deposit and 1st month's rent with no savings - especially, if, as is probable if you move to an area with better employment, your new rent is higher ? What if your former landlord keeps back some or all of your previous deposit ? It's no good saying that these things can be overcome if you're moving to a better job ....... who's going to wait (landlord or removals) for your 1st pay packet ? .... no-one! And no, not everyone has friends or family willing or able to step in and lend or gift you the money required. And no, not everyone has stuff they can sell to cover these costs (anything remotely valuable will have probably gone a long time ago).
The issue of retraining is perhaps even more difficult when you're on your uppers. As Artemis says, if you're in low paid work the govt sponsored retraining schemes are all but closed to you. Even if you find something - how do you get there if you can't afford transport costs ? How do you support your family in the "meantime" (the years between training, qualifying and actually getting a better paid job) if your bursary (if applicable) is less than the sum you're earning currently in your low paid job, never mind if you don't get a bursary at all! In similar past threads I've seen people roll out all the old clichés like you have to "speculate to accumulate" or "short term pain, long term gain" ..... which, in some circumstances, may very well be true but it is literally impossible to "speculate" at all for some. I've been in OP's position in the past, wanting desperately to improve my lot, but completely unable to take on any further debt as I wouldn't have been able to make repayments and would have no doubt ended up in even a worse position (things only improved for me when I met someone and stopped being a single parent but quite clearly I wouldn't advocate that solution, nor is it realistic for couples!). It isn't just about being able to afford fees, or finding a "free" scheme, but about living while you're doing it FFS!
This is NOT about being defeatist (or lazy, or feckless ) it's about being realistic. The infrastructure to support people in improving their lives - and ultimately, the country's economy - just isn't there any more for a significant number of people trapped in low paid jobs and awful accommodation. It isn't just about "moving" or "retraining" .... it's all the associated stuff you need in addition which enables you to achieve those goals: affordable childcare, affordable transport, accessible credit (and I don't mean Wonga etc), available jobs, decent affordable housing etc etc etc.
Most people accept that improving your life probably isn't going to be "easy" and are prepared to work hard to get what they need/want but there has to be a level playing field before we start inferring that poverty is always the fault of the impoverished - because it bloody well isn't. Some of the hardest working people I know are also the poorest. Look at the OP and her DH - 4 jobs between them - which she says wipes them out. Yet "we" are constantly fed the message that hard work is the answer to all ills - it isn't as simple as that, and in order for hard work to be effective, as in, providing a basic standard of living, a whole load of other stuff also needs to be in place. The people I've personally come across over the years who've moved from a very bad place to a more acceptable one have very rarely done so purely on the basis of hard work and compromise - there has always been an extra element of luck involved, e.g. a family member stepping in to offer free/subsidised accommodation, childcare or money, or a chance meeting with a neighbour who suggests a reciprocal childcare arrangement, or maybe an inheritence - that sort of thing, the sort of "leg up" which most people will not be able to access.
OP - I don't know what the answer is, and I'm sorry to have gone off on one on your thread. Reading about situations like yours though makes me so angry on your behalf. I don't know what THE answer is - but I do know that how you're forced to live isn't right. I think what everyone else has said about kids needing love more than "stuff" is absolutely true but I also appreciate it perhaps isn't quite as simple as that when you get to the stage of trying to feed them nutriciously, or ensuring they live in a home that's warm enough. There are so many aspects to being poor which maybe don't occur to people who've never been there - the grinding boredom, the restriction on your movements because you can't afford to travel further than your feet can carry you, the breakdown of friendships and wider family relationships because again you literally can't afford to get to people - and the resulting isolation - it just goes on and on and there's never a break from it, or light at the end of the tunnel. You are absolutely NOT scum though - you DO work hard and probably a damn sight harder than many of those who'd label and judge you for being "poor". Please please do not be ashamed to access any offers of help you think could benefit you - it is absolutely "freecycling". Stuff which would otherwise have languished in people's lofts for years or chucked into landfill. It's far better to use this stuff and you can then help others in turn by passing it on when you no longer have a use for it. I don't know what else to say OP - it seems so useless to say I hope things improve (when that goes without saying) after all the stuff I've written about how damn near impossible - or sod that, completely impossible it is for many to improve your life. I hope you can take some small comfort from knowing that despite the attitude and message that comes across in some sections of the media, and from some on here it has to be said, there are also a great many people who are absolutely disgusted and outraged that so many families have to live the way you do through absolutely no fault of their own.