In this Letter, we introduce four distinct classes of f-deformed photon-added nonlinear cat state. This would be achieved by recalling a nonlinear coherent states approach, as well as a particular class of Gilmore-Perelomov-type of SU(1,1) coherent state and a class of SU(2) coherent state. We then examine the role of photon addition and nonlinearity functions in the phase space structure and sub-Poissonianity of even, odd, and Yurke-Stoler nonlinear cat states. The effect of photon addition, which results in a π phase shift at the origin of the Wigner function toward negativity, interestingly enhances the nonclassicality by means of the Wigner function and Mandel parameter. Furthermore, owing to photon addition, we can observe a deformation in the Gaussian shape of the Wigner function, which may be found to be potentially useful in quantum noise reduction. Moreover, the deformation function shows a remarkable role in revealing the nonclassical behavior and can increase the depth and the domain of nonclassicality.