You see me once in June, twice in November and not at all in May. What am I?

I'm on the far right, and on the far left. Look either way, and I've moved farther away. No matter your center, I'm at the edge of your gaze

Note to all riddlers, here is a small clue. The title sometimes hints at what is hidden from view. In this little riddle, that word is 'WORD', A hint in plain sight, though rarely explored. Check other riddles; not all have these hints though. But those that have, may help ideas flow.

Use me before off, and people will leave our get agitated, Some creatures makes this sound, to the same affect, Quiz shows may name the button after me, bar two letters, To beat the clock, and you've beaten me, but remember to remove some letters.

I arrive with the moon, but I bring the storm. Pain is my harbinger; gore is my form. I sometimes wear wings and sometimes a cup. In fifty or sixty years my time’s up.