Sorry to hear that your horse has developed a cough. A cough can develop from a number of causes. The most common of which are upper respiratory tract infections or inflammation. I would consider having your veterinarian out to evaluate and auscult your horses lungs, and even perform a rebreathing exam as we can have lower respiratory tract infections, or even chronic airway disease that can lead to a cough. In which case antibiotics or other medications (Clenbuterol - a bronchiole dilator) may be recommended.
Depending on the cause, the cough tends to be one of the last clinical signs to develop but also one of the last to go away. After a respiratory infection I have had patients whose cough takes multiple weeks to resolve even though they clinically appear normal (except for the cough). Most viral infections (upper or lower respiratory) tend to resolve over 3-5 or 5-7 days, but again the cough may persist longer.
A few things you can do as an owner to help are to keep your horse in a well ventilated area, and wet down any hay or pelleted diet they are fed. This helps decrease the dust particles and other allergens that can stimulate or exacerbate an already inflamed pharynx of your horse. I also recommend removing the horse from work as the more we make their lungs work, the more inflamed they can become, making the cough that much worse. Also monitor for any excessive nasal discharge, or color changes to the nasal discharge. All horses should have a small amount of clear nasal discharge from the nasolacrimal ducts which drain the tears from the eyes. Also monitor you horses temperature daily (normal = 98.5 - 100.8F) for fever. This can be a signs of a viral infection or a bacterial pneumonia developing.
There really isn't many medications that you can give to help reduce the cough, that may not test by next week. You can try using a short course of Phenylbutazone as an anti-inflammatory. The majority of other medications which we can use to treat a cough generally have longer withdrawal times (guifenacin, Clenbuterol, Albuterol, ect.). Your veterinarian may prescribe a short course of antibiotics to help prevent a secondary bacterial infection.
I hope this has answered your question fully. If it hasn't or if there is anything else I can answer for you, please do not hesitate to ask.
With what you are describing, your horse may just have residual inflammation associated with a respiratory infection that sounds like it occurred a week ago. As he has been returned to work, he is now using the smaller airways that he doesn't use at rest. There is most likely a small amount of inflammation remaining. It is almost like when you or I have bronchitis, after the initial illness we begin to feel just fine, however, we have a lingering cough for an extended period of time. This is due to inflamed tissue being stimulated leading to the cough. Depending on the governing body USEF, FEI, ect., you can still use Bute at a certain dose up to a certain point before showing. Usually 48-72 hours before the show. You can often times use some antibiotics as well, but again I would check with the show, and what regulations they are following regarding medications. As far as Diphenhydramine (Benadryl), it has a longer half-life approximately 10 hours, and can remain detectable for upwards of 5-7 days. This medication is often a banned substance because it has moderate sedative effects as well. With an upcoming show, I would not recommend Benadryl or Hydroxyzine which has an even longer half-life.
Again, it appears that your horse has a residual cough most likely associated with a previous respiratory infection. Now that he is returning to work, the increased airflow and lower airway usage is stimulating residual inflammation. Typically they work out of this over a few days, however, I always recommend light work only as we don't want to over stress the respiratory tract and make things worse. With typical respiratory viral infections, the rule of thumb is 1 week rest for every day of fever. So it sounds as if your horse should be ready to return to some light work, and I would expect the cough to remain present but improve and become less frequent over the next few days. If not, it may be wise to have your veterinarian out to auscult the lower airways and perform a rebreathing exam to make sure there isn't a persistent low grade infection persisting.