Okay, he seems to have Seasonal Allergies.
(I am not a Doctor, this is just my opinion).
I have seasonal allergies or to just dust in the air and during windy days it is more irritating. I have been like this since I was a child.
So, my parents got me an air purifier for my room and the house.
But I grew up with pets etc.
But no carpet in my room etc.
I did not get allergy shots.
Nor allergy treatments etc.
And mainly because, my allergies was not and is not debilitating and I know and my parents knew, what irritated my symptoms.
As an adult, well I take Claritin. I do not like Zyrtec personally, because it makes me TIRED and GRUMPY.
Then I also have Asthma so I have an inhaler. But this was diagnosed and per my Doctor, since childhood.
The thing is: per seasonal allergies... you can see a specialist. And depending on how bad or debilitating his allergies are, then he can use prescriptions. Personally I never had to take the multiple shots per week. And my parents didn't want me to have to go through that.
AND also because, my allergies didn't bug me. Sure I sneezed/had runny nose/stuffed nose, etc. But it really was a no big deal to me. And my parents never went overboard in giving me things for it each night.
At a certain point, it is useful to do allergy testing. Especially if the allergy is debilitating or dangerous for the child/person. But I do not believe, that simple seasonal allergies, has to be dealt with with multiple shots every week. As a kid and even now, I am fine despite my seasonal allergies. And I just need to have a Kleenex box around.
As long as there is pollen and dust and especially on windy days, a person's seasonal allergy will get a tweaked. It will always happen.
And, the "cough" that you get because of it, is typically due to the "post-nasal drip" of the mucus going down a person's throat. This causes the "gag reflex" to then trigger and the person coughs.
Coughs occur because of: (1) post nasal drip, or (2) chest mucus, or (3) both.
As a child and now as an adult with those allergies, it really is not debilitating for me.
So I have not had to go overboard in taking medications for it. And I just live with it.
BUT if your child's allergies are debilitating or dangerous for him... then certainly prescriptions or targeted medicines, per his diagnosis, would help.
Try to keep your home, less dusty using an air purifier in his room and in rooms he frequents etc.
When he goes outside, change his clothes when he comes home or back inside, because, pollen and dust that is unseen by the naked eye, can still be clinging to his clothing and then it will irritate his allergies.