Someone has made a complaint about me to HR. It is completely false, but I'm worried that my employer's opinion of me will change. Can I ask for it to be struck off my record?

The definition of "personal data" is wide enough to include a complaint about you. You can make a written request to your employer to remove your personal data from its records on the ground that processing the data (which includes keeping it on its records) is causing or is likely to cause unwarranted substantial damage or distress. However, this is subject to certain limitations.