Do I need to call the doctor if my child has a cat scratch?

Yes, infection? Yes, kitten can sometimes cause cat-scratch disease which can show up in 3 - 10 days after the scratch. It looks like a red crusty bump with or without pus. Important to look for lymphnodes on the side of the scratch, at the elbow, underarm or neck, if on the leg, look around the groin. The lymph node can become tender and drain. Your child can also run a fever. Prognosis is good with antibotics.
Depends on cat. Any wound of the skin may allow entry of germs & produce infection. Good wound care is all that is needed unless you have additional concerns. Was the cat an unvacinated stray? Is there rabies in the region & could the cat have been exposed? Can the cat be cought & monitored? Was it a kitten that could cary cat scratch disease? A call to your doctor during regular hours is appropriate.