  We consider the formation of temporal localized structures or Kerr comb
generation in a microresonator with inhomogeneities. We show that the
introduction of even a small inhomogeneity in the injected beam widens the
stability region of localized solutions. The homoclinic snaking bifurcation
associated with the formation of localized structures and clusters of them with
decaying oscillatory tails is constructed. Furthermore, the inhomogeneity
allows not only to control the position of localized solutions, but strongly
affects their stability domains. In particular, a new stability domain of a
single peak localized structure appears outside of the region of multistability
between multiple peaks of localized states. We identify a regime of larger
detuning, where localized structures do not exhibit a snaking behavior. In this
regime, the effect of inhomogeneities on localized solutions is far more
complex: they can act either attracting or repelling. We identify the pitchfork
bifurcation responsible for this transition. Finally, we use a potential well
approach to determine the force exerted by the inhomogeneity and summarize with
a full analysis of the parameter regime where localized structures and
therefore Kerr comb generation exist and analyze how this regime changes in the
presence of an inhomogeneity.
