Can a day care provider charge for time that a child is not there?

I hope your son is feeling better.  Yours is a contract law question and it is legally permissible to charge under contract for daycare, schooling, etc.  even if the child is not there.  From the provider's perspective, they need to pay for staff and space whether or not a particular child shows up.  Parents can sometimes find "drop-in" care or pay-by-the-day but unless you had that kind of agreement you'd likely be responsible for normal weekly charges under the contract.  You may want to check the terms of your contract.