Well done for taking this step. There are a couple of things you can do to make things better. First claim UC credit as soon as you can, you can do it online. Any space explain that you are escaping Dv & have no other resources. After claim is verified, you can ask your advisor or Gp to refer you to a Dv support group/therapist or advisor ASAP. This person can then comb thru your finances to maximise. Are you claiming child benefit, you can do it online and ask them to backdate to the day you left. Also ask for UC to be calculated to the day you left.
If you are responsible for Council tax at new accomodation apply for all the relevant tax discounts online. Then when you get the bill and you still are left with only £100 after all bills, apply for something called discretionary council tax relief to cover remaining portion of council tax for rest of year. Every council has special fund like this, they give to pple in vulnerable circumstances. You are eligible due to no income no resources, no savings, caring for minors and Dv. They should cancel out your whole bill if you explain well.
About housing, once they send your housing benefit award letter, it will say how much they will pay, most councils work it in house and automatically pay the whole rent bill. If this does not happen and you have a rent balance, apply for something called housing discretionary fund/relief. All councils have it and it works like the council tax discretionary fund.
Councils also have hardship funds and may give supermarket vouchers, cash, gas top up, etc. Just work out your budget and see where you need help.You can apply online.
When school starts, approach the school and explain and ask for free school meal voucher, holiday voucher and afterschool club. Sometimes you can apply online. If you left all clothes & belongings behind you can ask for school uniform voucher, laptop fund etc. These bits help.
if your temporary accomodation is not furnished you can also ask the council for white goods & furniture. Always ask for help from council hardship team, housing officer, Dv advisor/support worker, Gp, school.
Also, some water companies have special tariffs and discounts for situations like this, look at their website or speak to someone on the phone. Remember to ask that any award/help is backdated to the day you left your Dv situatio.
Are you in london? If yes I can post up some example links.
Good luck,