Sorry to hear that you have had a strained voice for the past 6 months, and that you recently noticed a white lump with a vein running over it on your tonsil that has you worried. I am happy to give you some general information, and some different ideas as to what might be going on, but ultimately I am going to recommend that you make an appointment with an ENT or otolaryngologist
to get evaluated. They will be able to take a thorough history and examine you, which will let them make a much more focused and accurate differential diagnosis that I am able to speculate about. First off, there is a very low likelihood that the tonsillar lesion is related to your hoarseness
in any way, unless you have reflux that is causing your hoarseness and is also causing inflammation of your tonsil (although this would typically be erythematous (red) and not white like you describe). A white lesion on the tonsil could, as you accurately describe, be related to tonsiloliths (white are also called tonsil stones). It could also be a tonsillar cyst which can be whitish and is benign. Rarely would a tonsil cancer present in this way, but nonetheless an ENT would be able to examine you and tell you exactly what is going on. I wish you the best of luck.