H & L = High and Low. They’re the stop screws to stop the chain falling off either side. One for high gear (big ring), one for low (small ring).
Barrel adjuster is your indexing, and that will affect the shifting up and down.
Set H & L first so that the cage on either position will not allow the chain to drop off the side. It could be you’ve got them stopping it dropping off one of the rings or shifting up. Once set, leave these alone. They aren’t to do with shifting, just sets the limits.
Then crank the pedals, try to shift and adjust barrel adjuster if it’s struggling. Small turns either way until it seems to make a difference.