1//2 bed - Professional "single mum" 4yo twins

Hi!
I never thought I would be doing this but post-divorce I am starting to look at my options to move out with my kids of almost 4yo. They might not be all the time as we plan to do share custody.
I am finding very hard through Agencies to allow me to be with them sometimes even in houses that might be suitable for us within a reasonable one-person budget.
I work as a part-time B7 Occupational therapist for the NHS and combine with other sources of income from Independent work as an OT and commercial modeling.
This is very frustrating as I am a very responsible person and professional but the cost of living is just ridiculous expensive that you need 2 incomes to barely survive.
Anyway, I wanted to explore my options but of course worst case scenario I would continue sharing with their dad, so that I avoid being basically homeless.
Thanks for reading and do not hesitate to contact me if you know of a place that could be suitable for us. We are currently in Staines but I work in Wandsworth so anything south west would be ideal.