May I extend a double entry L visa to an X (extending it from the second entry)?

Certainly your school will go through a visa agency based in China for your visa application. So it is ok either on the first entry or the second. As I know, the processing in this way will cost much as the visa agency charges very expensive for their service. So. it is better for you to apply for a X visa in your home country by yourself. Or if possible, apply for it in HK if you do not want to go home. But may I know your nationality? because applicants from some countries are banned from applying China visa in HK.