Apparently, the Cauchy function means a solution of the ODE with initial conditions $x(s)=0$, $x'(s)=1$. In this case it is $x(t,s) = \frac12 \sinh (2t-2s)$, which wouldn't be too hard to find even if not given.
To build Green's function on $(0,2)$, we should start with $0$ at $0$, which any multiple of $x(t,0)$ does. Then, at a point $s$, change the derivative (but not the value) to eventually arrive at $x'(2)=0$. The change of derivative should be by $1$ so that the second derivative is Dirac delta at $s$. Thus,
$$g(t,s) = Ax(t,0) + x(t,s)$$
where $A$ is found from the condition $\partial g/\partial t=0$ at $t=2$. Compute:
$$A \cosh(4) + \cosh (4-2s) = 0$$
which determines $A$, and with it $g$.