Pokemon-go – Can’t Add Pokemon to Friendly Gym

The most likely cause of this is that all of the gym's slots are full, which means you will not be able to add another Pokémon until the gym is upgraded. Gyms can hold a number of Pokémon equal to the gym's level, so the gym in question is likely only level one, and its slot is already full. To increase a gym's level, you must win friendly "battles" (training) against the gym using only one Pokémon (as opposed to the usual six). Every time you beat a Pokémon at that gym, its prestige will raise, eventually leveling it up and adding another slot. At this point, you should be able to add a new Pokémon to the gym.

As Riley mentioned, the button will appear in the bottom left corner of the gym when there is an empty slot, and any Pokémon that you wish to place in that gym must be full health.

This error is related to connectivity. There are 3 variations of the issue:

1. Your defending count is desynchronized and you don't actually hold as many gyms as the screen shows. This issue should be fixed upon closing and restarting the app, viewing the Trainer screen by tapping your avatar's face in the lower left corner of the overworld view sometimes resynchronizes and fixes issues like this as well. If you still hold any gyms after resynchronizing, it should work unless it's (also) affected by the second variation of the problem.
2. Your server connection is unstable and you don't actually collect the rewards. When you tap the icon, your game sends a message to the server requesting the rewards, but the server never receives the request because of connectivity issues. This can be fixed by improving your connectivity. Try restarting your data connection by going into and out of flight mode or rebooting your phone, or find a place with WiFi. Once your connection is stable enough, if it still doesn't work, check for variations 1 and 3.
3. Your server connection is unstable, but you did in fact collect the rewards. This is similar to the second variation, only this time, the server received the request, but your phone didn't receive the confirmation. The coins have been credited to your account and the clock is running, but your game doesn't show either of these things because it didn't get the memo. When you try again, the server declines the request because the clock is running. As such, this variation is a visual bug only and will be fixed once you close and restart the app.