You cant hit me ;)

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Main Deck:
-3 Lapse of Certainty: This only buys you one turn. Also need space for my suggested inclusions:
-3 Forsake the Worldly: Enchantment and / or artifict hate is almost exclusively sideboard tech.
-4 Sunbeam Spellbomb: There's no real reason for lifegain here. Nothing triggers off yiu gaining life.
-3 Celestial Flare: There's better options for removal.
-1 Gideon of the Trials: With only one Gideon in the deck, his emblem is almost useless.
-3 Banisher Priest: Creature removal is heavy in Modern, plus there's better noncreature options.
+4 Howling Mine: You need the draw to guarantee Fogs in hand.
+4 Ghostly Prison: Slows down the number of creatures attacking you, allowing you to fog less often.
+4 Journey to Nowhere: Your Banisher Priest upgrade. Less mana and safe from creature removal.
+2 Day of Judgment: Going creatureless allows you to play cards like this without worry.
+2 Greater Auramancy: Protects your enchantments, but should be sideboard, sadly there's no space for them in the sideboard.
+1 Ethereal Haze: Your best 1 mana fog should be 4x.

Sideboard:
-1 Forsake the Worldly: There's more mana efficient replacements for this.
-1 Celestial Flare: Unconditional removal should be mainboard.
-1 Elixir of Immortality: You've got two main deck already, you don't need more.
-1 Orbs of Warding: Witchbane Orb costs 1 less and the damage reduction is less meaningful with Ghostly Prison in the deck.
-2 Day of Judgment: Things like this should be mainboard, especially since you effectively have no creatures.
+1 Witchbane Orb: Fogging is nice, but it doesn't stop spells targeting you. Leyline of Sanctity would be better though.
+2 Blessed Alliance: Your upgrade to Celestial Flare. More options is better.
+2 Disenchant or Sundering Growth or Fragmentize: Your upgrade to Forsake the Worldly. You want cheap mana answers to keep open fog mana.
+1 Greater Auramancy: Extra enchantment protection.

Don't think this works at all.

Fogs only delay attacks while the opposing army keeps on growing. That means at some point you need to Fog every turn until you can finally sweep the board. But this deck has only 2 mass removals spells, virtually no carddraw and no way to replay Fogs. Decks like this usually run Isochron Scepter. They are 3 dollars each, is that why you don't run them? The next best thing would be Howling Mines to constantly draw fogs, but it's also 3 dollars a pop. But without those, the deck just won't survive. It's a Turbo Fog deck without the "Turbo" part. You say you have card draw in those cycling spells and spellbombs, but that's not carddraw, you are just replacing cards, and 4 of the 7 cycling spells you have are the fogs you are searching for ... sorry, but 11 fogs without any way to draw more than one card per turn is a recipe for disaster.
I don't understand why this is mono-white either. IF you are really so budget-desperate that you can't afford compareably cheap but crucial staples like Isochron Scepter or Howling Mine, why then insist in mono white? Green has enough fog variants to fill an entire deck!

Other problems:
- It is beyond me why you would want to run a single copy of Approach of the Second Sun since you need at least two copies to make it work. This deck has no wincondition!!!

- Banisher Priest is completely useless in this deck! First, at some point you'll be forced to play Day of Judgment, which means you'll release whatever the Priest banished. But more importantly, it's the only creature you have, so the opponent will always have means to immediatedly kill it, since all his crature removal spells will sit in his hand, waiting for targets!

- You rely on cycling for carddraw but forgot to add cycling lands

- Lapse of Certainty is useless here as well, all it does is delaying the things that kills you one turn. It might be useful in a deck that kills the opponent in short order but this one here is a control deck that aims to drag the game forever.

Dedwards made good suggestions but I doubt that will be enough to fix all of those problems or even the main problem.