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In this study, we construct the abundant families of soliton solutions for the generalized (2+1)-dimensional Camassa–Holm–KP equation. This equation is used in tiny amplitude shallow water waves. The solutions are obtained in the form of Jacobi elliptic function solutions which are the soliton and solitary wave solutions. They are observed in the form of shock, bright, combined complex shock, singular and periodic waves, as well as rational solutions. Some techniques are applied to obtain the solutions with the help of Mathematica namely as; ϕ6-model expansion and a new modified extended direct algebraic method. In nonlinear dynamics, the acquired results are particularly helpful for studying and confirming the analytical solutions with numerical and experimental work. Additionally, the contour diagrams and three- and two-dimensional representations of the solutions are drawn for the various parameter values.