I did not furnish all the details because it would be too long, but I give some hints at the end of the answer.
I have used the formulae $:T^g: ~= ~:2(\partial c) b + c(\partial b):$ and $:\frac{1}{2}cT^g: ~= ~:bc \partial c:$, when there is an ambiguity in the calculus.
We begin by :
$$j_B = cT^m+:\frac{1}{2}:cT^g:+\frac{3}{2}\partial^2c=cT^m+:bc\partial c:+\frac{3}{2}\partial^2c \tag{4.3.3}$$
We have $T(z) = (T^m+ T^g)(z)$, we want to compute the OPE $T(z)j_B(0)$.
Note that $T^m$ has zero OPE with the ghost fields $c,b$ or $T^g$. Note that $c$ has holomorphic weight $-1$ and $\partial^2c$ has holomorphic weight $+1$
We have :
$$T(z)j_B(0) = T^m(z)c(0)T^m(0)+T^g(z)c(0)T^m(0) + T^g(z)c(0)T^g(0) \\+ T^g(z)\frac{3}{2}\partial^2c(0) \tag{1}$$
The first term is :
$$T^m(z)c(0)T^m(0)\sim [\frac{c^m}{2z^4} + \frac{2}{z^2}T^m(0)+\frac{1}{z}\partial T^m(0)]~c(0) \tag{2}$$
The second term term is :
$$T^g(z)c(0)T^m(0) \sim [\frac{-1}{z^2}c(0)+\frac{1}{z}\partial c(0)]~T^m(0) \tag{3}$$
The third term term is :
$T^g(z)c(0)T^g(0) =:2(\partial c(z)) b(z) + c(z)(\partial b(z)): :b(0)c(0) \partial c(0):\tag{4}$
The part concerning one contraction is :
$$\frac{1}{z^2}:b(0)c(0)\partial c(0) :+ \frac{1}{z}:\partial(b(0)c(0)\partial c(0)):\tag{4a}$$
The part concerning 2 contractions is :
$$-\frac{4c(0)}{z^4}+\frac{3\partial c(0)}{z^3} \tag{4b}$$
The fourth term term is :
$:T^g(z):\frac{3}{2}:\partial^2c(0)): = :2(\partial c) b + c(\partial b):\frac{3}{2}:\partial^2c(0):$, and this gives :
$$\frac{3}{2}[-\frac{6c(0)}{z^4}-\frac{2\partial c(0)}{z^3}+\frac{\partial^2c(0)}{z^2}+\frac{\partial^3c(0)}{z}]\tag{5}$$
Summing all the terms $(2), (3),(4a), (4b), (5)$, we get the desired result :
$$T(z) j_B(0) \sim \frac{ c^m - 26}{2z^4} c(0) + \frac{1}{z^2} j_B(0) + \frac{1}{z} \partial j_B(0) \tag{4.3.11}$$
Some hints :
The result $(5)$ is obtained by starting from :
$$:T^g(z)::c(w): = - \frac{1}{(z-w)^2} c(z) + \frac{2}{z-w} \partial c(z) \tag{6}$$ then deriving $2$ times relatively to $w$, and finally doing a Taylor expansion of $c(z), \partial c(z)$ around $w$, and finally putting $w=0$.
The results $4a$ and $4b$ are quite long and fastidious, you have to remember that, before doing one contraction or 2 contractions, you have to re-order the terms, and this may give a minus sign because of the anticommutation in the ordered product. For instance, if you have $:ab:~:cde:$, and you have a contraction $ac$ with a contraction $be$, you reorder by $acbed$, you have 2 transpositions, this will get a sign $(-1)^2 = 1$
