That depends IMO. Is the "mentor" really the "manager?" If someone with power over your performance evaluations and future at the company is swearing under their breath every time you do something wrong, that's a problem that HR maybe should be involved in.
HR is not there to solve your problems. HR is there to protect the company.
It's fine to be friendly with HR (some wonderful people work there), but remember who is paying them and whose interests they are supposed to look out for first. Definitely not yours.
HR wants all the info they can get from you. They will cheerfully listen to everything you tell them. And then they will do their job, which is to analyze the situation for potential risks around legal exposure, and take steps to mitigate those risks.
I would basically never bring HR in to a problem unless you are very confident that solving your problem and the goal of protecting the company are truly in alignment. Maybe retraining the manager is better for the company, but who knows. They can also solve their problem by removing the new employee who is being treated badly.