The impact that the COVID-19 pandemic has had, and will continue to have, on food security and child health is especially concerning. A rapid, Short Message Service (SMS) Maternal and Child Health survey was conducted in South Africa in June 2020 ( n = 3140), with a follow-up in July 2020 ( n = 2287). This was a national cross-sectional survey conducted among pregnant women and mothers registered with the MomConnect mhealth platform. Logistic regression was conducted to explore the associations between breastfeeding, maternal depressive symptoms, and hunger in the household. High breastfeeding initiation rates and the early introduction of other foods or mixed milk feeding were found. The prevalence of depressive symptoms in this survey sample was 26.95%, but there was no association between breastfeeding behaviour and depressive symptom scores (OR = 0.89; 95% CI: 0.63, 1.27). A positive correlation was found between not breastfeeding and not going to the health clinic. The odds of hungry mothers breastfeeding were significantly lower (OR = 0.66; p = 0.045). This result also holds in a multivariate framework, including covariates such as depressive symptoms, attendance of a PHC facility, and whether the infant was older than 3 months. Support for breastfeeding must include support, such as economic support, for breastfeeding mothers, to enable them to access nutritious diets. Mothers also need reassurance on the quality of their breastmilk and their ability to breastfeed and should be encouraged to continue to attend the health clinic regularly.