Noone is attacking you but it does seem like you don't want to see any other point of view other than your own.
There are ALWAYS choices. It's just a matter of what the other choice is and how appealing it is. You've made several choices which I believe has contributed to your current situation and it's a shame you won't consider any other thought process other than your own.
You CHOOSE to cook separate meals because you don't want your child to go hungry.
You COULD send your DD to another school which is closer but you choose not to because you fear she may get bullied and not get the same level of education.
You CAN apply for jobs you cannot do but you choose not to in case you get an interview 
You CHOOSE not to appeal the decision of the lone-parents advisor because you don't want the stress.
All these are choices you make. The latter two contribute to the fact you now have no money. The school thing makes it harder for you to be employable because you have even more restricted hours than you otherwise would have.
So the question remains....what do you want? Because if you are serious about wanting to work then you need to sort your life out because being blunt, no job will fit around you.
Yes it will mean your kids will have to make sacrifices, changes, be more responsible etc. But hey, they have to live within the means you can provide. And right now you are really doing them a disservice by not at the very least making the effort needed to claim the right benefits so they can live more comfortably.
Do you REALLY want to live like this for the next four years (or more)?