We say "have a runny nose" (not "have runny nose").

We say "have a runny nose" (not "have runny nose").

My child has a runny nose.

My child has runny nose.

Singular countable symptoms, such as "runny nose", are preceded by an article (a, an, the) or another determiner (this, that, my, your, his, etc.).

Mary has a runny nose and keeps coughing.

Having a runny nose is a typical flu symptom.

Peter went to school with his runny nose.

A partial list of symptom names that also take the article a/an:

• Allergy
• Backache (The article can be omitted in British English.)
• Cold
• Cough
• Cramp/Muscle Cramp (The article can be omitted in British English.)
• Earache (The article can be omitted in British English.)
• Fever (The article can be omitted in British English.)
• Headache
• Heart attack
• Sore throat
• Stomachache (The article can be omitted in British English.)
• Stroke
• Toothache (The article can be omitted in British English.)