Pagination of results from different sources merged by a unified scoring function
I think $F$ and its details are irrelevant, an appropriate comparison function that uses $F$ internally turns your problem into a regular sorting one. Then your problem is solved in $O(n)$ using a regular selection problem.

Find the element with rank $pk$ in $O(n)$ using a selection algorithm. Similarly find the element with rank $(p+1)k$. Then do a single linear scan over the array to find all elements that have a value between these two in $O(n)$.