Answer: There are numerous diseases that can infect begonias and cause spots of different types. It is very difficult to diagnose them long distance.
I would suggest you consult with your local county extension to obtain a specific diagnosis and based on that, determine how to proceed. If a chemical control is needed, they will have the most up to date information on what to use and how/when is best to apply it. They may appreciate a photo of the overall plant and close-ups of the discoloring, or a freshly cut sample kept cool and enclosed in a clear plastic bag.
In the meantime, avoid wetting the leaves when you water, do not handle the plant when it is wet, and make sure it has good air circulation. Remove affected leaves and dispose of them in the trash to try to limit reinfection. Good luck with your begonias!
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