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This study provides a scholarly examination of fundamental concepts within the field of group theory, specifically focusing on topics such as the wreath product and powerful <i>p</i>-groups. We examine the characteristics pertaining to the structure of the wreath product of cyclic <i>p</i>-groups, with a specific focus on the groups that are powerfully embedded within it. The primary discovery pertains to the construction of the powerful wreath product and the quasi-powerful wreath product. In this study, we establish that subgroups are powerful within the wreath product, specifically focusing on <i>p</i>-groups. The aforementioned outcome is derived from the assumption that <i>p</i> is a prime number and <i>W</i> is the standard wreath product of two nontrivial cyclic <i>p</i>-groups, denoted as <i>G</i> and <i>H</i>.