10D. Ice Cream

After the first melting step, the volumes become $3$ and $3$. Hannah licks the first ice cream, so the volumes become $1$ and $3$.

After the second melting step, the volumes become $0$ and $2$. Hannah licks the second ice cream, so both volumes become $0$.

At the start of the next second, all ice creams have volume $0$, so Hannah wins.

After the first melting step, the only ice cream has volume $0$.

Now every ice cream has volume strictly less than $k$, so Hannah loses before she can win.

After the first melting step, the volumes become $1$, $3$, and $3$. Hannah licks one of the ice creams with volume $3$.

After the second melting step, the volumes can become $0$, $0$, and $2$. Hannah licks the last ice cream with volume $2$, so all volumes become $0$.

At the start of the next second, Hannah wins.