In the absence of wind, current will influence the direction your boat lies at anchor. With a light or moderate current alone, the pool should not be greatly adversely affected. When wind and current are both present, there should not be a great effect on your boat and the pool if the wind and the current are running in roughly the same direction. A stronger current running from bow to stern will tend to elongate the pool and make it more oval. Mooring lines tied from the side tie straps to cleats or stanchions on the boat can help counteract this effect to some degree.
Most problems arise when the wind and the current are coming from opposite directions. In this case, the wind will tend to hold the boat pointed in one direction (toward the wind), while the current pushes the pool from the other direction into the stern of the boat. This is a situation where mooring lines and/or a bridle line are generally not effective. Swimmers pushing the pool against the current can sometimes help temporarily, but usually the pool will not take a circular shape until the current slackens or changes direction after a few hours.