Getting sick from the shot is a common misperception. In the next 48-72 hours, those people will mature their illness and get sick. Therefore, their sickness is a result of prior incubation of a virus, not the flu shot.
Children receive a split-dose vaccine.
The split-dose requires 2 doses, as your doctor indicated. Your child should receive that second dose. Therefore, vaccine can and should be received throughout the flu season. This season normally concludes in February or March. It is recommended to get the vaccine early.