Am I legally responsible for the back pay of discounted childcare?
I am prohibited from giving a legal opinion. The answer will be a a subjective determination for the court to make, based upon all the facts and circumstances involved. If in fact you aren't qualified for the discount, in order for the court to determine that you aren't liable for the underpayment, the court would need to find that it would be an injustice to enforce the policy and make you liable for the underpayment.
In some cases where a party has detrimentally relied on a promise made by another, the other party may be prevented from forcing another to perform their obligations. In order to succeed in such a defense, it must be shown that it was reasonable to rely on the promise, and such reliance led to harm that can only be prevented by enforcing the promise. For example, if you were aware or otherwise should have known that you were ineligible for the discount , it would typically not be found reasonable for you to rely on the misrepresentation otherwise.