Not all coughs are the same
I wil deal here with “simple coughs”, not with pneumonia
I will discus in short just 2 herbs which are readily avaiable in all homes.
Garlic and Thyme
Galic: [ can be given in in all forms]: excellent antinicrobial
Reaches very readily the lungs, as we all know as eating garlic cause your breath to smell very quickly!
Good for stuck mucus with a
choking,gagging cough!
It thins the mucus!
Black pepper in chickensoup may have the same effect

Thyme [koranit] excellent antimicrobial and anispasmodic, For excessive mucus, to help drying this up [runny nose, post nasal drip
As a tea, a tincture. or as a syrup.
Take very often some tea [every few hours, not just 3 times per day!]

Sambucus [I discuused this in the past, see, my WordPress] : excellent antiviral., even for prevemtion of Corona As a syrup with a very good taste, it may help to give thyme or garlic[[in small amounts] or other herbss to children