IS IT ALWAYS POSSIBLE TO COVER OR TOUCH UP A TATTOO?

In most cases, No!
A serious tattoo shop doesn’t cover or touch up any kind of tattoos.
Only the ones which are clearly horrific or made by amateurs, and only in the case of certain good result of the work.
It is also unethical to put hands over another artist work, so this is another reason why we avoid this kind of works.