This is an area in which I have some expertise and much experience as I have been making colonial reproduction furniture and restoring antiques for decades along with making teak joinery for even longer.
Ross's first answer is the correct and easiest one to implement.
Oxalic acid is one of the strongest of the organic acids and is a reducing agent that does not damage wood fibers. This is why it is the woodworker's preferred wood bleaching agent. It is readily available as the operative ingredient in Kaboom, Barkeeper's Friend and ZUD, all of which can be found inexpensively in your local supermarket. Make a paste of the powders or spray the Kaboom and apply to the stained area only. Soak for 30 minutes, wipe off and the rinse thoroughly with water. Repeat as needed. If you get it too light, a quick and light touch up wipe with Minwax red mahogany stain will get you a close color match.
A couple of coats of high quality tung oil will eliminate your water staining problems and will last for years. It is easy to apply, self hardening and very durable.