I think it is unreasonable to request a deposit AND a months pay before the start of care. The deposit IS your confirmation of their abiding by the contract and commencing care on the set date. They can pay the first month of care just prior to the first day of care.
If money is tight you still need to be responsible. It seems unwise to take a deposit and spend that money before the child starts. Even if the deposit is non-refundable it is technically supposed to be the fees for the set period of care (typically the first or final weeks of care). That money SHOULD be put in the bank and be left there until those weeks of care, to pay yourself. Spending it now, just leaves you moneyless then.
If the family does pay you for the first month of care DO NOT spend ANY of that money until it is YOURS. That money is ONLY for the period of care...if you spend it and they do not use that period of care then in many cases you need to return it. It will depend on your contract of course, but it isn't your money to spend before you work those days.