My left breast is aching no pain I had thyroid taken out am on Synthroid (thyroxine) would it cause it?

No. Synthroid (thyroxine) will not cause your breast to be painful, especially on only one side. See your doctor for an eval and physical exam.
Get breast checked. If you're on a correct dose of synthroid, (thyroxine) there's no connection. If you are a female with breast pain or tenderness, definitely have it evaluated by your doctor.