Single parent housing benefit situation

I’m a single mother with 3 children aged 6, 5 and 14 months. I’ve recently been told that my housing benefit will be cut by £120 a week. My 14 month old is currently under a consultant at hospital for breathing/lung problems, my 6 year has behavioural problems and is under a physiatrist to try and get to the root of the problem and i’m at my wits end with worry. Ive lived in London my whole life, all my family and friends are here, as is my kids father who sees them regularly and my housing officer has told me they will only provide a house for me in Manchester or Bradford. I told the officer i just can’t move. I suffered with depression two years ago, and still have down times where my family helps me out. There is also my kids schools to think about ect. I’ve never known anywhere but London, and that they’re effectively making me and my kids homeless.

He rudely told me ‘you’ll be homeless but your kids wont cause we’ll take them off you’. I have no idea what to do, they told me i could keep my money if i got a 16 hour-a-week job and that they would pay 75% of my childcare. Its just not possible to even get a job like that, i’ve looked everywhere! Let a lone pay the extra £250 a month childcare I have to make up (£1000 a month child care) or juggle the 6 weeks holidays.  And as if the rub salt into a raw wound, the woman before me smugly told me that she’s been allowed to keep her place and have all of her rent paid which is more expensive than mine just because she ‘played the racial card’ – as she put it – because she has two mixed-race babies. I'm furious that my kids should suffer, its a vicious circle that i can’t escape.  Is there ANYTHING i can do?