Usually when you enter the hole, there will be a brief cutscene where Kirby climbs down a ladder. Having a powerup above your head or in your mouth will cause Kirby to start falling immediately (left). It's possible to get past the Sawyer at the bottom, but it's pretty precise. Drill can be shot downward, breaking all the blocks at once. Throw away the powerup, get fire, and navigate to Shard 1. For both powerup swallows, you can do it while falling off a ledge to save a bit of time.
In the Dedede section, the primary concern is those crushing hammer machines. There are six in total, and ideally you alternate between waiting for them to fall and getting in front of the cycle. It's possible to get four early cycles, but it's really risky and not worth it. The whole time, there is a conveyor belt running to the left, so it's best to jump as much as possible. As shown in the clips, the best places to jump are enemies and "doorways." For the third clip, it's much better to jump over the Turbites than to hammer them. Finally, there is a Sawyer at the very end that's possible to jump over.
In this room, there are conveyer belts going both directions. It's faster to use fire on the red conveyer belts moving to the left. For the first I³ I like to use fire to get past it without taking damage. My visual cue is to fire right at the enemy's shadow. If done quickly enough, the fire should be a short one so that it ends around the time the I³ hits you. You can also get behind the I³ with a slide (middle), but you'll always take damage and it's more precise from my experience. At this point, remove fire and grab the cutter enemy. Unlike many of the previous combinations, Cutter is swallowed so that you can make use of headbutting. Finally, combine with the turbite on the ceiling. Note the small green conveyer belt optimizations in this section.
The miniboss fight can be a little finicky at times. As you enter, jump and start using the powerup, breaking the cage up top. As you fall, hit the Burnis either by turning around or swinging your lightsaber. The latter is more consistent, though slower because the animation is still going by the time you land. Jump and turn around to hit the Burnis three more times, and then take out your powerup and throw it for the remaining two hits. When the powerup is active, it takes a while to remove, so press down to put away the lightsaber. Afterwards, get fire from one of the Magoos that spawn, grab Shard 2, and leave. If the Magoo on the left just dropped into the lava as you threw the powerup, go get the one on the right since it'll spawn sooner.
The following room has five crushing platforms. Essentially, the only one you can get a quick cycle on is the fourth one, which involves using fire to get through the third one quicker. This saves about 8sec over the slower cycle, plus the pattern of the Shotzos in the background is nicer.
In the final room, you want to combine for fire sword using the fire powerup you already have. There are three cutter enemies, so keep fire until you reach the third one (left). After maneuvering around enemies, grab Shard 3 on the floor with the Scarfies. Afterwards, jump up and beat the crushing electric platforms. When jumping up to that floor, jump into the middle of the "grate" to avoid the bubble the Mahall shoots out. The slide off the ledge is crucial for getting past the platform. Afterwards, the Zoos enemies will try to throw sparks at you, but these can be headbutted out of the way.
If you die in one of the last two rooms, the best backup for HR would be to get lightsaber. There are a couple of spark enemies, which you can combine in the same way on the Fishbone.