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This is normally caused by too much nitrogen in the soil and not enough phosphorus. You may not be able to salvage this batch, but in the future, add bone meal to the soil when planting root vegetables.

Radishes can be stored in the refrigerator but are best used within two weeks of harvesting.

You can flood the area with water to dilute any remaining fertilizer in the soil and then add some phosphorus, like bone meal. This will help the plant produce more root (radish) growth.

This article will help with planting radish: https://www.gardeningknowhow.com/edible/vegetables/radish/tips-how-plant-radish.htm

It is likely slugs or snails. They feed at night and will leave holes in the leaves and little trace of their visit other than that. This article will help:
https://www.gardeningknowhow.com/plant-problems/pests/facts-about-slugs-and-how-to-kill-garden-slugs.htm

https://www.gardeningknowhow.com/plant-problems/pests/organic-snail-control-how-to-control-garden-snails.htm

If the damage is to the roots (bulb) themselves, this article may find helpful: https://www.gardeningknowhow.com/edible/vegetables/cabbage/control-cabbage-maggot.htm

As for foliage damage, there are several kinds of caterpillars that feed on the leaf tops. The standard treatment for almost any caterpillar is Bacillus thuringiensis, also known as Bt. You can also try neem oil. This is usually effective on those kinds of pests, as well as many others.