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Those are mushrooms! They will not, usually, harm the plants, as they only consume decaying, and dead matter. It is unclear what type that they are, as they are still very young, so please have caution around pets or children.

You may remove them, or you can leave them. They do serve a beneficial function to the soil, and I usually leave them to do so. Otherwise, everything looks healthy, and looks like they are happy where they are.

This article will help you in caring for these plants: https://www.gardeningknowhow.com/edible/vegetables/radish/tips-how-plant-radish.htm

Radishes like cool weather, but it needs to be warm and wet enough for them to fill out before the weather really heats up. Changes in temps are the likely cause.

There can be a few reasons for this and this article will help you access the care.
https://www.gardeningknowhow.com/edible/vegetables/radish/radish-does-not-form-bulbs.htm

This, usually, indicates a lack of phosphorus in the soil. It could also indicate that there is too much nitrogen, which is common with most potting soils.

This can be offset by adding more phosphorus, and potassium, as long as the soil pH is not neutral or alkaline already.

This article will offer more information: https://www.gardeningknowhow.com/edible/vegetables/radish/radish-does-not-form-bulbs.htm

This is most often caused by too much nitrogen in the soil. Soil for root plants, such as these, will need quite a bit more phosphorus, and much less nitrogen.

Here is an article that will help: https://www.gardeningknowhow.com/edible/vegetables/radish/radish-does-not-form-bulbs.htm

They are probably going to flower and seed. This is caused by heat and longer days. The only way to offset this is to plant earlier in the year so that you can get a harvest before bolting, unfortunately.

Here is an article to help you to prevent this from happening in the future:

https://www.gardeningknowhow.com/edible/vegetables/radish/radishes-bolting.htm