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This article may help:

http://www.gardeningknowhow.com/edible/vegetables/beets/different-beet-varieties.htm

Mike, without knowing more about your soil and other growing conditions, I can only guess that the problem is one or both of these factors:
1. soil tilth is the physical condition of the soil, including is it loose and friable with good organic matter content, or is it tight and compacted with impenetrable clayey texture. Root vegetables in particular need good tilth. Soil improvements may be needed to create better conditions for you root veggies to grow below soil grade.
2. Soil nutrient availability and balance. Fertilization, products and timing are important. Fast release chemical Nitrogen fertilizer or manures can cause excessive vegetative growth at the expense of the root/tuber development. A balanced, slow release organic fertilizer is best. Bone meal or fish bone meal provides Phosphorus which is needed for root crop development. Adequate soil organic matter is important, which is best provided by incorporating compost into the soil before planting. After planting, a layer of mulch is the best you can do.

This article on beets has some general information that is applicable to other root vegetables too:

https://www.gardeningknowhow.com/edible/vegetables/beets/beets-small-or-deformed.htm

Application rates are usually per 1,000 sq ft. I'm a big fan of soil testing to find out exact fertilizer needs. Many soils are naturally high in phosphorus and adding more with a balanced fertilizer like 10-10-10 can actually slowly increase phosphorus to harmful levels - harmful to plant growth, not human health. Every state has a cooperative extension service based at your land grant Agriculture school. They do testing for a nominal fee and allow you to designate what you are growing. Results include pH, phosphorus, and potash levels (sometimes more nutrients like magnesium) and how much NPK is needed annually. If phos or potash are low, corrective measures are listed. It is the same if pH is out of acceptable range for the crop. Beets need compost and/or manure added to the soil annually to keep them sweet tasting. Since you have probably already planted, add a top-dressing with 2 inches of compost.

You can cut off the damaged areas of the root and wash all harvest.

https://www.gardeningknowhow.com/plant-problems/pests/animals/garden-mouse-control.htm

This collections of articles will give you everything that you need to know on the subject of beets, how to grow, harvest, and other useful tips: https://www.gardeningknowhow.com/edible/vegetables/beets

Actually, they are right at a point where they should be planted for a second harvest of the year. Leaving them might allow them to mature fully for a good harvest in a couple of months.

This website will give you a good chart for when to plant things in your area: https://www.ufseeds.com/learning/planting-schedules/Zone-6-Planting-Calendar

This article will give you more information on growing beets: https://www.gardeningknowhow.com/edible/vegetables/beets/growing-beets.htm