Hello and hope you are doing well.
It’s unlikely that coughing or gagging can cause a tear in the throat. The pain could be due to the constant coughing causing an inflamed throat. Also, the cause for the constant coughing like a throat infection or a post nasal drip can also cause an inflamed throat and pain. To find a cure you need to consult your primary care physician for proper assessment. In the meantime for relief from your symptoms you could do warm salt gargles and take OTC antihistamines.
Hope this helped and do keep us posted.