  We present Cosmological models with modified Chaplygin gas (MCG) in the
framework of Horava-Lifshitz (HL) theory of gravity both with and without
detailed balance. The equation of state (EOS) for a MCG contains three unknown
parameters namely, $A$, $\alpha$, $B$. The allowed values of some of these
parameters of the EOS are determined using the recent astrophysical and
cosmological observational data. Using observational data from $H(z)-z$, BAO
peak parameter, CMB shift parameter we study cosmologies in detailed-balance
and beyond detailed-balance scenario. In this paper we take up the beyond
detailed-balance scenario in totality and contribution of dark radiation in the
case of detailed-balance scenario on the parameters of the EOS. We explore the
effect of dark radiation on the whole range the of effective neutrino parameter
to constrain matter contributing parameter $B$ in both the detailed-balance and
the beyond-detailed balance scenario. It has been observed that greater the
dark radiation less the matter contribution in the MCG in both the scenario
considered here. In order to check the validity of beyond detailed balance
scenario we plot supernovae magnitudes ($\mu$) with redshift of Union2 data and
then the variation of state parameter with redshift is studied. It has been
observed that beyond detailed balance scenario is equally suitable in HL
gravity with MCG.
