$k$th Largest Algorithm in a range $[k, k+c]$
Step 1. Use QuickSelect to find the $k$th largest number in the array.  Call it $x$.

Step 2. Use QuickSelect to find the $k+c$th largest number in the array.  Call it $y$.

Step 3. Find all elements in the array that are at least $x$ and at most $y$.  This can be done by simply iterating over the array and comparing each element to $x$ and $y$.

Each step can be done in $O(n)$ time.  Therefore, the total running time is $O(n)+O(n)+O(n) = O(n)$ time.