I overstayed in Hong Kong ten years ago and was sent back to the Mainland by Hong Kong. Does this history affect my application for one-way permit now?

Have an impact. Because one-way permit has an important basis: whether you have been punished for entry and exit in the past history. If you usually overstay (OS days), you will generally be exempted from the fine, or you will be given a visa with a small fine for a few days (according to the length of your overstay), so that you can go back to the mainland through customs normally, and there will be no problem, which will have little impact on your one-way permit application (yes, but not serious). But if you say "sent back to the mainland", the problem is even more serious, because it involves your criminal responsibility in entry and exit issues. I suggest you consult the staff of the Immigration Bureau (you can not give your name, or as your friend, ask about the consequences of this situation or the impact on the one-way permit). Or visit the website of the Immigration Department and consult through the website: http://www.immd.gov.hk/zhtml/home.htm.