In this paper, we adapt and extend B. Birnir, G. Ponce, and N. Svanstedt result related with local ill-posedness for the Cauchy problem to the mKdV equation in [5] to the generalized $p$-Gardner equation. We prove that the Cauchy problem for the generalized $2$-Gardner equation is ill-posed in the Sobolev space $H^s(\mathbb{R})$ for $s < -\frac12$.