Grate the ginger and turmeric into a 3-quart saucepan with a coarse microplane. Add the honey and water, and stir to combine.
Bring the mixture up to a bare simmer over low heat, stirring occasionally. Turn off the heat, cover the saucepan and let it steep for 10 minutes.
Line a fine-mesh strainer with a layer of cheesecloth and place it over a bowl, then strain the mixture from the saucepan into the bowl. Discard the solids and return the strained liquid to the saucepan.
Return the saucepan to the stove over low heat. Add the apple cider, heat until piping hot, then ladle into mugs and serve.