In the present work, we characterize Lorentzian para-Kenmotsu (briefly, $LP$-Kenmotsu) manifolds with a quarter-symmetric non-metric connection (briefly, QSNM-connection) $\widehat\nabla$ satisfying certain $\ddot \phi$-semisymmetric conditions admitting Ricci solitions. At the end of the paper, a 3-dimensional example of $LP$-Kenmotsu manifolds with a connection $\widehat\nabla$ is given to verify some results of the present paper.