— Maria Ferrer (@QueenVisionEnt) July 8, 2014
A raspy voice can have several reasons. For example, it could signal vocal imbalance, effects of a cold or allergies, or even by a pathological vocal condition. From your questions it sounds as if the raspy condition of your student’s voice has been present for quite a while now. In any case you should have your student see an ENT to make sure you are medically safe to continue with voice lessons (if you haven’t already done so).
If the singer’s voice is healthy the rasp might be the result of too much tension. There are many ways to approach this problem, but since you said that you have tried quite a few we suppose that this singer’s voice is a hard case. Therefore, it might be necessary to isolate the student’s falsetto for a while. For this purpose you could use “ooh” like in moon or “eeh” like in “we” on a legato scale, on a short interval (e.g. a second back and forth). Start above the first bridge and move this scale up and down in half steps. The rasp should then disappear. Once this begins working you can alternate between falsetto and pharyngeal exercises to connect the voice again.