Hi,
While removing the phlegm you won’t have injured the baby hence the blood. It is not advisable to put your fingers inside baby’s mouth.
While trying to clean nose too, we can accidentally injure the block, so use saline nasal drops to clean.
Next Steps
Continue the Medications given by the doctor. Give plenty warm liquids to drink.
Health Tips
If continuous nose bleed or bleeding from other site with fever, meet your doctor.