Defects play a crucial role in understanding the physical and mechanical behavior of materials. In this study, the fracture problem of an infinite one-dimensional hexagonal piezoelectric quasicrystal material matrix containing secondary asymmetric straight cracks with lip-shaped pores is investigated. A defect mechanics model of secondary asymmetric cracks with lip-shaped pores is constructed for the first time. Utilizing conformal transformation technology, a conformal transformation formula from an infinite region containing secondary asymmetric cracks at the lip on the physical plane to the outer region of the unit circle is built. Using the complex variable method, analytical expressions for the field intensity factor and energy release rate at the crack tip are obtained. Under given conditions, these analytical results can be simplified into solutions for other defect models, such as secondary single cracks at the lip and secondary symmetric cracks at the lip. At the same time, they can also degenerate into the solutions of classical Griffith cracks and lip cracks without secondary cracks. Numerical examples reveal the effects of defect size, particularly the lip height and crack length, on the field intensity factor and energy release rate. The results show that increasing the length of both sides of the crack promotes crack propagation, while increasing the height of the lip inhibits crack propagation. These findings are consistent with the conclusions drawn from theoretical analysis. When the length of the secondary crack on one side of the lip is zero, as the height of the lip increases, the stress intensity factor and energy release rate at the crack tip on the other side first increase to a peak and then gradually decrease, eventually stabilizing at a constant level. The research results presented in this paper can contribute to the development of a theoretical framework for material fracture mechanics and provide technical assistance for nondestructive testing, reliability design, and optimization of piezoelectric quasicrystal material equipment and components.