Yang-Baxter sigma models, proposed by Klimcik and Delduc-Magro-Vicedo, have been recognized as a powerful framework for studying integrable deformations of two dimensional non-linear sigma models. In this short article, as an important generalization, we review a non-integrable sigma model in the Yang-Baxter sigma model approach based on [arXiv:1406.2249]. In particular, we discuss a family of deformations of the 5D Sasaki-Einstein manifold T1,1, instead of the standard deformations of the 5-sphere S5. For this purpose, we first describe a novel construction of T1,1 as a supercoset, and provide a physical interpretation of this construction from viewpoint of the dual Klebanov-Witten field theory. Secondly, we consider a 3-parameter deformation of T1,1 by using classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). The resulting metric and NS-NS two-form completely agree with the ones previously obtained via TsT (T-dual - shift - T-dual) transformations, and contain the Lunin-Maldacena background as a special case. Our result indicates that what we refer to as the gravity/CYBE(Classical Yang-Baxter Equation) correspondence can be extended beyond integrable cosets.