After the firing of a neuron, the sodium and potassium concentration differences vanish.
It requires some time for cell to actively transport the ions in and out to re-establish the balance.
Does the HH model incorporate this effect?
HH doesn't "count ions", it pretends the reservoirs are infinite.
Specifically, the "reversal potential" for each ion species is a constant. This constant is calculated from the Goldman Hodgkins Katz equation which uses the concentrations on either side of the membrane to calculate the reversal potential given the concentration gradient.
We could hypothetically keep this a function of the individual species concentrations and add a current to the model to represent the pump, but you would gain little in predictive power and much in computational complexity.