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The gun was meant to make it possible to eliminate the enemies, but since enemies in current build has virtually next to no threat it is next to no use, unfortunately.
Also the health upgrade increases your maximum health by certain amount, but as the same reason as the gun item, it's not really necessary to beat the game.
Those two are obviously my mistake on game design, and I am going to revisit the game design in later updates when I have more time to rethink & explore more on the game's idea.