If my employer asks why I was late to work, do I have to disclose my medical condition was the cause

Hi, I am a teacher at a high school in CT. By contract I must be into work at 7:10 am. I have a medical condition that sometimes prevents me from getting to my job by 7:10. I was late perhaps 4-6 times in the span of 5 months. I was never more than 20 minutes beyond the 7:10 contract time; usually 5-10 minutes beyond, but late nonetheless. I was asked by an administrator in my building why I was late and I did not feel comfortable stating the fact that I had a medical condition that was making me occasionally late, so I said it was a day care issue, as I was a bit embarrassed and also wanted to keep my medical information private. I am now being disciplined for "lying to an administrator" as I later disclosed my condition when I was in a pre-disciplinary meeting about my lateness and they are putting a letter in my file. Do I have any recourse to argue why I did not disclose my medical condition at that time?