Magnetic field in the air gap of a transformer core
Your formulas are correct. You copied those from a good source :-) As for the Hc, just note that the flux lines go round the entire magnetic circuit of core and gap, so that the flux density, which is B, will be equal in core and gap. Hence:

B = μ0 * Hg = μ0 * μr * Hc


Thus:

Hc = Hg / μr


As μr for a ferrite or iron core may be well over 100 or 1000, Hc is correspondingly much smaller than Hg. Most often you can neglect Hc * Lc compared to Hg * Lg, but this depends on your μr. In any case, the sum of all H * L parts equals your Ampere-Turns:

4 * 800 = Hc * 0.132 + Hg * 0.01

        = Hg * (0.132 / μr + 0.01)


You see that the term 0.132/μr may be overshadowed by 0.01 only if μr is in the thousands.

Now you will be able to compute your H in the air gap given the μr of your core.

As for the energy, that could not be stored in the core as noted by some commenters: the energy per unit of volume is about H * B / 2.
So if there is no air gap, all energy is stored in the core. But because of μr being large, and B must be below saturation at e.g. 0.3 Tesla, H should not be very large, so you should not apply many turns and/or many amperes. On the other hand, if you apply lots of Ampere-Turns and get into saturation, the HB product and the stored amount of energy will rise as if you had an (expensive) air core. 

With air gap, use the same math as above to arrive at the amounts of energy stored in core and gap.