Thanks for your recent question.  It is possible that another Master Gardener may have already responded but in the event that this was not the case, I will attempt to answer your question.

 There are two primary organisms that could be doing damage to your radishes:  slugs or root maggots.  I am attaching two pics showing the differences in damage to your radishes associated with these two causes.

To control the slugs, cut plastic pots or food containers into rings about 4 inches high. Place these around your plants, pressing the edge about an inch below the surface.  It has also been suggested that spreading diatomaceous earth around radishes can prevent slugs from reaching the plant.

Root maggots are associated with a small fly that emerges in the spring.  These flies then lay eggs in the soil next to your radishes.  When these eggs hatch, the resulting maggots burrow into your radishes.  These maggots subsequently develop into pupae that are in the soil next to the radishes.  These pupae spend the winter in this state.  In the spring, these pupae develop in the adult fly and the process starts all over again.  Other crops that may be affected by root maggots include broccoli, radishes, cabbage, cauliflower, turnips, carrots, and onions. They are more common during cool wet springs.  If you're describing what happened last summer and you're planning a radish crop this year, you may want to invest in a floating row cover to protect your crop by early to mid-May which is when the flies are starting to lay eggs. It may be a bit too late this year. Since it was not much of a cool wet spring this year, you may be in luck but think about a row cover next May.  It might also be a good idea to do your radish planting next season in a different place than you did this year.  Be sure to remove all your radishes this fall to minimize the spread of maggots into next season.

 Here are a couple of sites that will give you further information

http://www.extension.umn.edu/garden/yard-garden/vegetables/growing-carrots-and-root-vegetables/

http://homeguides.sfgate.com/prevent-slugs-radishes-74621.html

https://www.extension.umn.edu/garden/insects/find/root-maggots-in-home-gardens/

 https://www.gardeningknowhow.com/plant-problems/pests/insects/root-maggots.htm

 https://www.planetnatural.com/pest-problem-solver/garden-pests/root-maggot-control/

 Good luck