Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I'm sorry to hear that your fellow has a cough and I understand that you are concerned about him.
A cough simply means he is experiencing airway irritation.
A cough can be related to heart disease, including heartworm, lung disease secondary to a viral, bacterial, fungal or parasitic infection or tracheal irritation due to a bacterial, viral or fungal infection.
In a young dog infectious disease is by far the most common cause. Of those kennel cough or infectious tracheobronchitis is one of the most common causes. If you know that he has been around other dogs (parks, boarding kennel, out on walks) then a viral/bacterial infection called kennel cough would be high on my list of possible problems.
Even if he has been vaccinated we cannot vaccinate against all of the agents that cause kennel cough and some dogs will still get sick even with the vaccine. The good news is that dogs that are vaccinated usually don't get as sick as dogs that haven't been vaccinated.
As long as he is still able to eat and drinking well, isn't lethargic, isn't running a fever (rectal temperature greater than 103F), and he doesn't have a green or yellow nasal or eye discharge then you can try taking a watch and see approach and using an over the counter cough suppressant.
Make sure he gets plenty of fluids, lots of rest as the more active he is then more she will cough, and eats well. Add warm water or low salt chicken broth to his food. The better hydrated he is the better he will feel and the thinner his respiratory secretions will be.
You can take him into the bathroom with you when you take a hot bath or shower so he can breathe the steam which is soothing to his airways and thins mucous, making it easier for him to breathe. You can also use a humidifier in the room he sleeps in,
His cough should slowly resolve over the next 2 to 3 weeks. Most of the time these dogs feel pretty well so it's a matter of keeping them quiet and suppressing the cough if we can.
If the cough is very bad you can use human cough/cold medications with Guaifenesin (an expectorant) and Dextromethorphan (a cough suppressant) only such as Benylin or Robitussin DM can help. Give 1 teaspoon per 20 to 25 pounds every 6 to 8 hours. I understand that you have tried Robitussin but are having a hard time giving it to him. You can mix it with a little canned food, meat baby food or yogurt and try giving it that way, or if that doesn't work he may find Benylin less bitter tasting and be more agreeable about taking that.
If he is running a fever this is almost certainly a contagious infection and if he is running a fever and is lethargic then an examination is definitely warranted as pneumonia may have developed.
If his cough persists then this is likely something more serious than a minor infection and he does need to be examined.
If your regular veterinarian is beyond your means financially now your other option is to contact your local shelter or humane society and see if they run a low cost clinic or know of one in your area.
If not here is a web-page that lists some that may be able to help:
Other sources of help can be found here: http://speakingforspot.com/?p=Financial%20Assistance%20for%20Veterinary%20Care
Best of luck with your fellow, please let me know if you have any further questions.