If the email you want to add to your HackerRank account belongs to a different HackerRank account with no activity, you can delete the account using the Delete Account feature on the Account Settings page.
Add the new email ID to the Email Addresses section on the settings page. You will receive an email from HackerRank to confirm your access of the ID.
Once you have confirmed your email, the entry will show up as verified on the settings page. You will also have an option to "Make primary". Click on that option.