HR can be damaged during his "slap" attack (left video) simply by holding the sword out and jumping over the outstretched hand. A simple strat for the "clap" shown in the middle can get three hits on HR. It's possible to squeeze in four attacks into one clap with the strat in the video on the right. There is a small space where Kirby can stand: too far to the left and the sword doesn't hit the arm, and too far to the right causes Kirby to get hit. My visual cue is to align the end of Kirby's left foot with the green and black part of the walkway. I've found that getting the fourth hit is easier when you jump.
During the second phase, you can hold the fire sword vertically and hit HR's "nose." After three hits, HR will fly up into the sky and try to drop on top of you. The goal is to manipulate HR into the middle so he becomes vulnerable again sooner. Once he lands, do another three hits to finish him off. It's possible to throw the double-powerup, but I don't think it's worth it since it can hit one of its attacks instead.
In this phase, HR has three attacks: rockets, claws, and one big rocket. For rockets, the attack probably won't hit you. The claws will attack in front of HR, so stand farther to the right while still attacking. The big rocket, will get hit by the sword as long as Kirby stands still.