My rates that would cover this situation are what I use when a child is integrating. It is $15 for a morning or afternoon if they are picked before lunch or arrive after lunch. It includes one snack. If they add lunch to either session it is an extra $5.

The fact he is there half a day but no lunch if you told the mom $20 a day that is probably a good rate. And it isn't something that is going to run long term.