Children have mild COVID-19 disease compared to adults, and up to one-third are asymptomatic. The immunological basis for this difference is unclear and data on the cellular immune response in children with SARS-CoV-2 exposure and infection are limited. In a previous case study of two parents with PCR-confirmed symptomatic SARS-CoV-2 infection and their three SARS-CoV-2 PCR-negative children, we showed that all family members had a cellular immune response characterised by striking changes in the frequency of innate immune cells over time, particularly among the children. Here, we report acute and convalescent innate immune responses of 48 children and 70 adults infected with, or exposed to, SARS-CoV-2. We found that clinically mild SARS-CoV-2 infection in children was characterised by low proportions of all three circulating subsets of monocytes (classical, intermediate, non-classical), dendritic cells and natural killer cells during the acute phase, whereas SARS-CoV-2-infected adults showed reductions in the non-classical monocyte fraction only. Increased proportions of CD63+ activated neutrophils during the acute phase were an additional feature unique to SARS-CoV-2 infected children. Both children and adults exposed to SARS-CoV-2 but negative on PCR testing had increased proportions of low-density immature neutrophils that were observed out to 7 weeks post exposure. This study describes a role for the innate immune response during SARS-CoV-2 infection in children and suggests that exposure to SARS-CoV-2 induces a change in the immune response irrespective of evidence of active viral infection. Our study provides insights to explain differences in disease severity between children and adults.