I know you need the childcare for your sanity, but could you cut it down to one day a fortnight to save a little money? If you go to a playgroup you could enquire about setting up a babysitting club for if you need more childfree time than that. A lot of mums are in the same position with little money so you may find you can make an arrangement with some of them. I know how tough it can be coping with no family to help out, but if it comes to it you may have to look at dropping the childcare because it is possible to live without it, unlike your other major bills and food.

Where do you shop? I save a fortune by avoiding the major supermarkets and get everything from Aldi. Can you contact your phone provider and see if you can get on a better plan (mobile and landline). Could you cope without a landline?
Also look through your bank statement for the last month (or keep a record of all spending for the next month) and identify any areas that could be trimmed. Things like take-away coffee, magazines and trips to playcentres can be replaced with a thermos, a library pass and trips to the park.

I would recommend contacting centrelink and talking through the payments you are getting to make sure there is not something you are missing financially. It seems strange your payments were dropped when they said they wouldn't be.
I hope you find something you can do to help improve your situation. It sounds like you are doing it really tough and I'm sure your MIL removing half your possessions hasn't helped you any

ETA I hope you don't mind me answering without being a single mum, I used to work in banking and have a bit of experience with budgeting as a result.