For the persistent cough please try nebulizer with baking soda. ( 1 tea spoon of baking soda for the cup of water ).
I hope it will help your kid.
Mama to Many (TN) on 12/02/2022

Our family has had the terrible flu that has been all over the country.
We are mostly better, but the cough lingers (and relapse is common.)
I have been drinking vinegar tea and that has helped some. But this morning, I squeezed the juice of one lemon into a mug and added hot water and that seemed to be more effective for this cough.
My 11 year old has had a relentless cough and, after 2 weeks, was not really improving. I started using the nebulizer on him yesterday with 2 teaspoons 1% hydrogen peroxide 4-5 times a day. To make 1% hydrogen peroxide, I mixed up a jar with 1/4 cup of 3% hydrogen peroxide and 1/2 cup of distilled water. I then use 2 teaspoons of this in the nebulizer as often as desired. His cough today has been noticeably less. I am using hydrogen peroxide from the drug store in the brown bottle.
I also used that same hydrogen peroxide in the ears (straight 3%, not diluted) for anyone whose ears felt stuffy. This flu has come with some ear and sinus pain after the first week, many of us have found.
We have also been using Nyquil to help us sleep. (Just like the commercial promised back in the 80's.) It seems that returning to normal activity and not enough sleep are the recipe for relapse.
This flu has also had a seriously painful sore throat. My son asked for a throat spray, but the store was out. I made my own - 2 Tablespoons of honey and 2 Tablespoons of warm water and 1/4 teaspoon peppermint extract (not essential oil) I put that in a little spray bottle. It is tasty and helpful. Of course, gargling salt water is ideal, but a yummy spray is a better sell than gargling. :)
Take care everyone!
~Mama to Many~