Public users sees "Raise this request on behalf of"
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.