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When I run the same model, is it possible to have lower theoretical mean in second order approximation than in first order approximation?
I thought it should be higher due to constant (positive) risk adjustment.

For example, if I run second order approximation, I see "correction" term(let's say A) below "constant"(B) in policy function.
What I understand is that this correction term is related to the risk(i.e. variance of shocks).
But "A" has negative value, not positive value...

Anyway, I can see that "B" - "A(negative value)" = steady state value in first order approximation.

But I still don't understand why the theoretical mean is lower in the second order approximation..

There is nothing that tells us that Jensen's Inequality effects at second order can only go into one direction. Thus, it is not guaranteed that means are always higher when accounting for uncertainty. There are economic effects that can go in the opposite direction.