  In this paper, we study a projectable Ho\v{r}ava-Lifshitz cosmology without
the detailed balance condition minimally coupled to a non-linear self-coupling
scalar field. In the minisuperspace framework, the super Hamiltonian of the
presented model is constructed by means of which, some classical solutions for
scale factor and scalar field are obtained. Since these solutions exhibit
various types of singularities, we came up with the quantization of the model
in the context of the Wheeler-DeWitt approach of quantum cosmology. The
resulting quantum wave functions are then used to investigate the possibility
of the avoidance of classical singularities due to quantum effects which show
themselves important near these singularities.
