The moon is what causes all the trouble for The Man-Moth.
The moon has been used so often as a symbol in literature that it's hard to say anything new about it. It can represent madness and the dark side of human nature, but it can also represent fantasy and dreams. Interestingly, the moon's symbolic nature in this poem is fluid, its meaning dependent upon who is observing it at the time.
Line 2: Technically this line isn't about the moon, but the light from it, but that's close enough in our book. Specifically, the light is "battered." Is that deep fried? Beaten up? Either way, it's a metaphor used to describe the quality of the light itself. More than that, it's setting the scene for us, telling us it's nighttime and that there may be something sinister about this part of town. Sounds like the perfect place for a superhero to show up, even if he's a failed one.
Line 6: We see the moon, but the man doesn't, so it's not so much the moon itself but the man's chosen ignorance of the moon that makes it symbolic here. More than that, this moon is a "she." And while that might be how lots of writers describe it, we still have a case of personification on our hands. In this case, the moon represents enlightenment or inspiration. The moon is spilling its light all over the place, but the man just doesn't see it. It's kind of like when you spend ten minutes looking for your keys and they've been in your hand the whole time.
Line 11: Here again it's the moon's effect on someone else that makes it a symbol. The Man-Moth sees the moon and he is inspired to greatness. The poem is pretty direct here when it says "the moon looks rather different to him."
Line 14: The Man-Moth doesn't see a boring old rock in the sky, but a hole. It's a doorway to greater things.
Line 20: The word "moon" isn't used in this line, but (in a weird, extended metaphor) "that round clean opening." It sounds lovely doesn't it? Like a big, bright hobbit hole in the sky. If the grass is greener on the other side of the fence, then the sky must be more beautiful on the other side of that opening.