You operate a business and under your terms of business you expect to be paid - they are failing to meet that part of the contract.
Chasing payments is a real pain for any business. I used to work in an industry where clients had a lot of money, and it was sometimes because they didn't pay their bills very promptly... some people will just hold on to invoices for as long as possible. Poorer people tended to pay up quicker - anyone else find that?
Your client being on a 'money problem' is not your proble... it is theirs. If she has debt issues then she can ask you if you will consider varying your terms, so that she can pay a day or two later than previously agreed. You don't have to accept such a variation to the contract.
As your client is paying in arrears, perhaps at the beginning of next month, they could pay EXTRA, so that there is at least a weeks amount of money paid to you in advance (call it a deposit if you like) so that when they next fail to pay on time, you can use that deposit.
I would do the following:
Talk to the parent asking them to pay you by a specific date. Tell them you will produce an invoice for them showing exactly what they owe.
Produce an invoice for the client, showing all outstanding payments to date, requesting payment in full by {DATE}. If payment is not recieved, contract to provide childcare will be deemed to have been terminated by the parent, and childcare will thus no longer be provided, and the parent will owe the outstanding amount plus amount due under termination of contracted (notice period).
While it may appear harsh, you can't really provide parents with Free Childcare. They have to pay for the service if they want it to be provided. If they have difficulties in paying they can talk with you about the situation.
Business Link have some information regarding Payment Terms, though you should note that as you are requesting payments from individuals rather than from companies, the Payment of Commercial Debts (Interest) Act won't apply.