I do this more for cocoa or other drink powders that tend to clump/not dissolve easily. Mostly cocoa powder when mixing it with soy milk before microwaving.

But first I put the powder in the bottom of the mug. (If using any sugar add that and stir it through, taking the chance to crush out any lumps between the back of the spoon and the side of the mug. The rough granules both help to break lumps up mechanically and also the dispersion/dissolvability helps the liquid permeate the dry powder.) Then add a small amount of the hot water/milk/liquid and stir it into a thick smooth paste, continuing to crush out any lumps. Keep adding small amounts and stirring until it’s thin and smooth, then add the rest of the hot water/milk/liquid.

Maybe if you do that first then chuck the teabag in it might still steep like normal. (I add the milk immediately and leave the teabag in because I’m not waiting around to forget about the tea.)