If the white stuff is fungus it is doubtful that it is what is killing the plants. Are the roots ok? When my plants suddenly wither and die it is either because voles have bitten off the base of the plant or eaten most of the roots - or - the soil didn't drain well enough and the water logged conditions caused the roots to rot.

I'd say they drowned. That inch of water a week is a good rule for lawns which have far more plants per square foot than gardens do..especially mulched gardens. Some lawngrasses need more,some less..some on hillsides or near lots of trees might need twice that amount. Flower borders that lie downhill from a lawn could either be getting too much(if the rain is strong) or too little if the rains are gentle sprinkles.
So many variables but the clue is that thread fungus. It's not a cause of death but an indicator that your plants were weak and dying. The fungus is natures garbage disposal. Draw off that mulch and get some sunlight and air to the soil. Test it for drainage before deciding what to plant next year in that spot.