The time complexity for finding the kth smallest number in a min-heap Suppose that $k &lt; \sqrt n$, what is the time complexity to find the $k_{th}$ smallest number in a min-heap?

I thought that we can remove the root element for k times and each time we apply heapify? So it should be $O(k \log n)$ but the answer is $O (k \log k)$. What did I do wrong?
There is an $O(k)$ algorithm [1].

[1] Frederickson, G. N. (1993). An optimal algorithm for selection in a min-heap. Information and Computation, 104(2), 197-214.