Conceivably have you ever had a common cold? Peradventure did you eliminate the infection? Obviously you got over it! You can't cure a common cold itself, yet your body generally develops resistance to a specific cold virus within several weeks. That is called being cured by your personal immune system!

HPV is similar because HPV is just an additional viral infection. And you will generally develop resistance to HPV. However, HPV virus is more capable at evading your immune system compared to the cold virus viruses. Moreover you need to work harder to develop resistance to HPV.
https://www.openlearning.com/u/renwick-qmln5h/blog/HowToCureHpv/