Is it a cory? Like I said, I couldn't really see it...
But yes, a 10 gallon is perfect for a betta and a shoal of Cories. You should get at least 5 of them. Cories are very sensitive to water quality, so that's why you'll need to cycle the tank. Taking everything out and doing 100% water change once a week will get to be a huge pain (take my word on this, it's what I'm doing right now
), and your I don't think Cories won't do well with the constant change.
Since fish constantly release ammonia through waste, it builds up and can become quite lethal to fish in high amounts. Cycling is allowing beneficial bacteria to bloom in your tank. These take ammonia and convert them to nitrites, which are still pretty dangerous. Then, in the next stage, another type of bacteria blooms which converts nitrites to nirates, which are okay to have as long as you do the regular water changes. In a cycled ten gallon tank, you'll only need to change about 25% of the water per week.