Some babies develop allergies to food products which may present with facial rash, coughing or lots of regurgitation or vomiting, resulting in crying. Breast milk is the best form of nutrition for the baby as it is natural and is clean and available day or night. The mother's body usually filters out any impurities and allergens, but sometimes these can still cause rashes.
Formula fed babies seem to present with some rashes which may be related to the dairy component but could be any number of additives from vitamins to pre and probiotics or anything else used to stabilise the powder. If there are any signs that this might be happening (like poor growth or bloody or greeny bowel motions or lots of reflux) check with your health provider (midwife, plunket nurse or your doctor).
There are scheduled visits to the doctor in NZ at 6w, 3m, 5m, 15m, 4y and 11y. The doctor may send for some tests or make some suggestions to try alternative methods of feeding or finally see an allergy specialist. Unfortunately at this young age the babies' immune system is immature and any testing may be inconclusive. Testing only after 1y is recommended, and in the meantime consult with your doctor for further advice.