  The mass spectrum of $1+1$-dimensional $\mathrm{SU}(N)$ gauge theory coupled
to a Majorana fermion in the adjoint representation has been studied in the
large $N$ limit using Light-Cone Quantization. Here we extend this approach to
theories with small values of $N$, exhibiting explicit results for $N=2, 3$,
and $4$. In the context of Discretized Light-Cone Quantization, we develop a
procedure based on the Cayley-Hamilton theorem for determining which states of
the large $N$ theory become null at finite $N$. For the low-lying bound states,
we find that the squared masses divided by $g^2 N$, where $g$ is the gauge
coupling, have very weak dependence on $N$. The coefficients of the $1/N^2$
corrections to their large $N$ values are surprisingly small. When the adjoint
fermion is massless, we observe exact degeneracies that we explain in terms of
a Kac-Moody algebra construction and charge conjugation symmetry. When the
squared mass of the adjoint fermion is tuned to $g^2 N / \pi$, we find evidence
that the spectrum exhibits boson-fermion degeneracies, in agreement with the
supersymmetry of the model at any value of $N$.
