= Solution
The <mean-field approximation> replaces
$$
S_iS_j=m(S_i+S_j)-m^2+(S_i-m)(S_j-m)
$$
by its first three terms. Since every site has $q$ neighbours,
$$
\sum_{\langle ij\rangle}S_iS_j\simeq qm\sum_iS_i-\frac{Nq}{2}m^2.
$$
The sites then decouple, and their three possible weights sum to
$$
1+e^{-\beta g+\beta(Jqm+B)}+e^{-\beta g-\beta(Jqm+B)}
=1+2\kappa\cosh\!\bigl(\beta(Jqm+B)\bigr).
$$
Hence $c_1=1$, $c_2=2$, and
$$
Z=e^{-\beta NJqm^2/2}
\left[1+2\kappa\cosh\!\bigl(\beta(Jqm+B)\bigr)\right]^N.
$$
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