I notice you say it is New Zealand spinach (Tetragonia tetragonioides), which is actually very different from regular spinach (Spinacia oleracea). Common names are sometimes misleading. From looking at your picture it definitely looks like an advanced case of downy mildew. Some key clues that it is a downy mildew or a powdery mildew are the white powder-like accumulations on the leaf surface. This can then turn into lesion-like patches (I have also seen this on peas with powdery mildew).
Unfortunately it is hard to get rid of downy mildew once it has taken hold. As David mentioned, you can avoid to spray the upper parts of the plants with water, which will reduce spread. Instead water at the base of the plant. You can also try hand[spritzing the plants with a bicarbonate solution (baking soda). Put four tablespoons of baking soda in one gallon of water, along with a few drops of dish soap to help it adhere to the plants. If you apply this at regular intervals it can reduce the growth of the downy mildew (but be sure to wash your leaves well before eating). If the plants are in a very moist or shady spot, consider planting them in a sunnier, dryer spot next time to avoid the downy mildew returning.
Incidentally, it also looks like you have a chewing herbivore and a leaf miner attacking this plant. You might consider planting it at a different time next year to avoid some of the pests that are hitting it right now.