Here is an $O(k\log k)$ algorithm at this SO answer, slightly improved.
Create toVisit, a collection of nodes that contains the nodes which we will traverse next. This is initially just the root node.
Let counter c = 1.
While c < k:
Remove the smallest node from toVisit
Insert that node's children in the given min-heap to toVisit
Return the smallest node from to toVisit
toVisit can be implemented in a min-heap or balanced BST/B+ tree or skip list in which the insertion and removal of the smallest node takes $O(\log m)$ time, where $m$ the number of nodes. Assuming the degree of original min-heap is $O(1)$, there were at most $O(k)$ elements in
toVisit since there have been at most $k$ insertions.