There is no cure for HPV. Similarly there is no cure for the common cold or the flue. There is now a vaccination for the flue, HPV, but not the common cold, with varying effectivenesses for each. Viruses are not easy to cure. A cure for the common cold would make millions, perhaps billions, but that hasn't managed to get us there. Viruses are tough buggers.
Our bodies have dealt viruses throughout evolution. HPV and HSV are sequestered by the body in most people and controlled. For HPV that may mean the bodies effectively get rid of it, and for HSV it may mean you are no longer contagious. But none of this is clear according to the evidence. It is hinted at, not proven.
The evidence points to virtually all sexually active adults getting HPV. Just as we all get the common cold. It is normal to have HPV. It is unusual to not have it. HSV (herpes) is similar in that more adults have one form or the other than don't have it. The treatment for HPV is for the warts that some strains cause. Most strains do not cause warts. Some strains cause cervical cancer and thus are dangerous, but are in the minority. Pap smears help catch possible cancer as early as possible.
Likely there are many varieties of virus present in our bodies. We have evolved to live with them. There are many varieties of bacteria home to our bodies. We don't live in "clean" bodies. We host many other life forms within us. That is natural and normal.