In this article, we present a task and motion planning method for retrieving a target object from clutter using a robotic manipulator. We consider dense and cluttered environments where some objects must be removed in order to retrieve the target without collisions. To ensure a successful execution, the interplay between task planning ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">what to remove in what order</i> ) and motion planning ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">how to remove</i> ) is crucial. Thus, the task and motion planning approach combining a symbolic task planner and a geometric motion planner becomes one of the major paradigms in manipulation planning. However, motion planning in dense clutter often leads to frequent failures, so repetitive task replanning is inevitable. Although symbolic task planners are general and domain-independent, they do not scale; so we need an efficient task planner specialized for dense clutter for fast completion of tasks. We propose a polynomial-time task planner for object manipulation in clutter that can be combined with any motion planner. We aim to optimize the number of pick-and-place actions which often determines the efficiency of object manipulation tasks. We consider common situations that could occur in clutter: 1) all object locations are known, 2) some hidden objects are revealed while relocating some front objects, and 3) the target is hidden until some objects are removed. Our method is shown to reduce the number of pick-and-place actions compared to baseline methods (e.g., at least 28.0% of reduction in a known static environment with 20 objects). We also deploy the proposed method to two physical robots with vision systems to show that our method can solve real-world problems.