See this earlier MO question. The union of n disks (represented in terms of its boundary arcs) can be constructed in $O(n\log n)$ time. So construct the union of the $n$ disks you are given, and separately construct the union of the $n+1$ disks including the one you want covered. The given disk is covered iff if and only if the two unions are the same.
See this earlier MO question. The union of n disks (represented in terms of its boundary arcs) can be constructed in $O(n\log n)$ time. So construct the union of the $n$ disks you are given, separately construct the union of the $n+1$ disks including the one you want covered. The given disk is covered iff and only if the two unions are the same.