I touched the mark and did my penalty turn — do I have to go back and round the mark again?
No. Touching a mark breaks Rule 31, and Rule 44.1 lets you clear it with a One-Turn Penalty. Rule 44.2 says what that is: the turn made promptly, in the same direction, including one tack and one gybe. That is the whole requirement — nothing in Rule 44 mentions the mark. Once the turn is complete the penalty is complete, and if the circle happens to carry you round the mark again, that isn't a foul either. It just costs you boats.
<iframe width="560" height="315" src="https://www.youtube.com/embed/tkrWFLNmV_M" title="Sailing Kiwi" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
<p><a href="https://sailing.kiwi.nz/v/tkrWFLNmV_M/">Source: Sailing Kiwi</a></p>