TT OPE for bc ghosts and massless fermions

While calculating T(z) T(w) OPE for bc ghosts and massless fermions I came across a weird result.

I could get the central charge correct and also the T(w)/(z-w)^2 term correct but the other terms which remain does not add up to T(w) as expected.

In massless fermion case:

T = :ψ ∂ ψ:

T(z) T(w) = 1/4/(z-w)^4 + T(w)/(z-w)^2 + ( 1/2 ψ ∂² ψ + ∂ ψ ∂ ψ )/(z-w)

But,
∂ T = ψ ∂² ψ + ∂ ψ ∂ ψ

So, I cannot account for the 1/2 factor in front of ψ ∂² ψ .

In bc ghost case :

T = : 2 ∂c b + c ∂b :

T(z) T(w) = 1/4/(z-w)^4 + T(w)/(z-w)^2 + ( 4 ∂²c b + 3 ∂c ∂b - c ∂²b)/(z-w)

But,
∂T = 2 ∂²c b + 3 ∂c ∂b + c ∂²b

This means I have some terms like

T(z) T(w) = 1/4/(z-w)^4 + T(w)/(z-w)^2 + ( 4 ∂²c b + 3 ∂c ∂b - c ∂²b)/(z-w)
= 1/4/(z-w)^4 + T(w)/(z-w)^2 + [∂ T + 2( ∂²c b - c ∂²b)]/(z-w)
= 1/4/(z-w)^4 + T(w)/(z-w)^2 + ∂ [ T + 2 ( ∂c b - c ∂b) ]/(z-w)

I have no idea why the term ( ∂c b - c ∂b) should be zero.

So the problem is:
1. Whether it is correct to directly differentiate expression for T(z) to get the ∂T(z) ?
2. Is there a way to discard the total derivative using EOM or Belinfante tensor ..?

I have no idea still.