Problem
When public user sees Raise this request on behalf of and you would like to remove it
Solution
The dropdown would only be visible if you were using a service desk agent as the Public User which should is not safe, since all anonymous users will have agent privileges
Change your public user to a non-agent (i.e.) not Jira-software-user or jira-servicedesk-agent and just a user with customer privilege.
This would be solved.