Fluid mechanics concerns the mechanisms of liquids, gases and plasmas and the forces on them. We aim to investigate a \((2+1)\) -dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics in this paper. With the help of the Painlevé analysis, we find that the above system has Painlevé-integrable property. A set of the auto-Bäcklund transformations and some solutions are derived by the virtue of the truncated Painlevé method. We obtain certain bilinear forms via some seed solutions. According to the mentioned bilinear form, we derive the multiple-soliton solutions on some nonzero backgrounds. Based on the soliton solutions and conjugation transformations, the higher-order breather solutions on certain nonzero backgrounds have been obtained. Via some conjugation transformations, hybrid solutions formed from the breathers and solitons on certain nonzero backgrounds have been derived. We also graphically show the interactions between those solitons and breathers.
