In the US - at least at the institutions I'm familiar with - English proficiency is assessed by the graduate school, not the department. If you do not pass the minimum standards set by the university, the department cannot admit you.
If you do pass, your application is forwarded to the department, who may or may not see your scores. However - especially for equity reasons - many people are happy to abide by the thresholds that have been set for them.
If your score is below the admittance threshold, there is nothing you can really do except take the test again.
If your score is low, but passing, I would first ask the graduate admissions office whether the committee will even see your scores. If not, you're good. If so, I might very carefully write something about your score explaining your source of stress (if extraordinary - grad school is stressful) and that you were sick, if there is a box for "explain external factors."
If there's no spot for a cover letter in the application, then the one you send will most likely not be seen anyway.