The picture shown below is either growing out of my Rose of Sharon tree or another tree beginning – I really don’t want more trees where this one is – what is it – what do I do about it.
Its difficult to identify the leaves on the stem in your photograph. It looks like it could be a sucker from your Rose of Sharon, Hibiscus syriacus. Rose of Sharon occasionally sucker, particularly if they are planted too deep so that the root flare is buried (where the stem flares out at the base of the plant).
Pruning Rose of Sharon should be done when the plant is dormant, like late fall or early spring before the buds appear. Unwanted suckers can be pruned out at any time. Your problem stem looks damaged and should be cut off at its base.
If you think the leaves on this stem do not look like Rose of Sharon then I would still recommend cutting it out, as it is growing too close to your existing shrub.