Avoid homelessness for mum &2 kids

I have had to put aside all shame and embarrassment in doing this as I do not know what else to do. ... not easy.
I am a lone parent who works really hard to look after my 2 children.  Unfortunately, the house we currently renting has been put on the market and due to unforeseen circumstances (had no income for a period of time), we find ourselves in a difficult position to look for another rental property. However, following a terrible period in 2013, the effects of which are affecting me now as i i have to pay 6 months in advance rental fee or have a guarantor, both of which i do not have. It is frustrating as I can afford the rent but do not have the deposit and money that is currently required. I do not qualify for social housing and really need a boost to pay the deposit. 

I would be so grateful for any help towards and thank you in advance for your messages and any contributions.